Atmospheric Circulation without Presupposed Global Geometry

Cyclostrophic Sign Degeneracy, Stationary-Planar Mechanisms, and an Assumption-Explicit Program of Model Comparison

Abstract

The familiar association between pressure systems and preferred directions of atmospheric circulation is commonly interpreted through equations written for a rotating spherical surface. This interpretation is strongly developed mathematically and has extensive predictive applications, but the interpretation should not be confused with the observations themselves. This paper asks what may be inferred from atmospheric circulation without presupposing either spherical geometry or surface rotation.

Five principal findings emerge. First, cyclostrophic balance determines the magnitude of tangential velocity but not its clockwise or counterclockwise sign. This sign degeneracy applies to vortices in which an inward pressure-gradient force supplies centripetal acceleration; pure cyclostrophic balance cannot independently maintain steady circular flow around a pressure maximum. Second, pressure gradients, convection, baroclinic torque, shear, angular-momentum transport, and moving boundaries can generate organized circulation without planetary rotation. Third, a motionless planar model can, in principle, reproduce regional or domain-wide directional preferences if it contains an independently specified sign-selection mechanism, such as background atmospheric vorticity, differential boundary motion, spatially varying torque, or a velocity-dependent transverse field. Fourth, existing publications purporting to test “flat-Earth meteorology” evaluate particular nonrotating disk implementations rather than the complete set of stationary-planar models. Fifth, mathematical constructibility does not establish empirical adequacy. Competing models must be compared using common observations, explicit energy and angular-momentum budgets, independently measured forcing terms, and preregistered out-of-sample predictions.

The paper therefore proposes neither acceptance nor dismissal of a stationary-planar atmosphere. It establishes a model-neutral framework under which rotating-surface and stationary-surface explanations can be formulated, quantified, and tested on equal evidentiary terms.

Keywords: atmospheric circulation; cyclostrophic balance; gradient wind; planar atmosphere; model comparison; vorticity; angular momentum; baroclinic torque; boundary-driven flow

I. Introduction

Atmospheric observations and atmospheric interpretations are not identical. Instruments measure pressure, temperature, humidity, wind velocity, acceleration, optical phase shifts, and other local quantities. A theoretical model then relates those measurements to a larger geometry, coordinate system, and causal mechanism. A pressure map, for example, is not itself a demonstration of the mechanism that curves the wind. Likewise, a measured rotational signal establishes rotation relative to an instrument’s reference system, but further analysis is required to determine what physical component is rotating and relative to what frame.

Atmospheric science already uses a hierarchy of models rather than one indivisible representation. Simplified models isolate individual balances, while increasingly comprehensive models combine radiation, convection, fluid motion, surface exchange, topography, and other processes. Model hierarchies are valuable precisely because no single model is identical to the physical atmosphere.¹ An assumption-explicit inquiry should therefore distinguish three questions:

What circulation is observed?
What mathematical terms reproduce that circulation?
What physical mechanism produces those terms?

This distinction permits examination of a rotating-surface model, a stationary surface beneath a moving atmosphere, a stationary planar domain with moving boundaries, and a stationary domain subject to an external field without assuming in advance which interpretation is correct.

The central argument of this paper is narrow. The observed tendency of pressure systems to develop organized circulation does not, by itself, uniquely determine global geometry or surface motion. At the same time, an alternative model does not become empirically adequate merely because equations can be written for it. Each candidate must identify its force source, energy supply, boundary conditions, angular-momentum budget, and distinctive predictions.

II. Assumption-Explicit Equations of Motion

Consider a compressible fluid in a coordinate system that is initially left unspecified. Its momentum equation may be written schematically as:

Du/Dt = −(1/ρ)∇p + g + F_b + F_v

Here, u is fluid velocity, p is pressure, ρ is density, g represents gravitational or buoyancy-related acceleration, F_b represents any additional body force, and F_v represents viscous or turbulent stresses.

This expression does not initially require a sphere, a plane, a rotating surface, or a stationary surface. Those properties enter when the geometry, coordinate transformation, body forces, and boundary conditions are specified.

In a coordinate system fixed to a rotating surface, rotational terms may be introduced into the equations. In a stationary-surface model, those terms would be absent unless replaced by a physically specified force, moving medium, or boundary interaction. The observational question is therefore not simply whether a term of the mathematical form

a_T = K (z-axis unit vector × u)

fits the data. Different models might contain the same mathematical form while assigning it different physical origins. The discriminating question is whether K is independently derivable from measured surface motion, atmospheric background motion, an external medium, electromagnetic interaction, or another source.

The vorticity equation further demonstrates that rotation can be generated by several mechanisms. In schematic form:

Dω/Dt = (ω · ∇)u − ω(∇ · u) + (∇ρ × ∇p)/ρ² + ∇ × F_b + viscous terms

Vorticity can consequently be stretched, tilted, transported, created by misaligned pressure and density gradients, injected by boundaries, or generated by the curl of an external body force. Atmospheric research has specifically documented baroclinic production of vorticity in mesoscale and convective flows.² Rotation of the underlying surface is therefore one possible contributor to organized vorticity, but it is not the only mathematical source of vorticity.

III. What Cyclostrophic Balance Does and Does Not Establish

For a steady, approximately axisymmetric vortex, the cyclostrophic radial balance is commonly expressed as:

v_θ²/r = (1/ρ)(∂p/∂r)

This uses a sign convention in which pressure increases outward from a low-pressure center. Cyclostrophic models are used for compact, rapidly rotating vortices in which the radial pressure-gradient force and inward centripetal acceleration dominate the balance.³

Two conclusions follow.

First, because v_θ appears as a square, the radial equation admits:

v_θ = +√[(r/ρ)(∂p/∂r)]

and:

v_θ = −√[(r/ρ)(∂p/∂r)]

The equation determines speed but not clockwise or counterclockwise direction. Direction must be selected by initial vorticity, environmental shear, boundary stress, imposed torque, background circulation, or another term not contained in the elementary radial balance.

Second, the same equation does not provide a pure cyclostrophic explanation of a steady vortex centered on a pressure maximum. Around a pressure maximum, the pressure-gradient force points outward, while circular motion requires net inward acceleration. Atmospheric studies of balanced flow accordingly recognize that pure cyclostrophic winds are not available for high-pressure configurations; anticyclonic gradient-wind solutions require an additional inward term in the force balance.⁴

This qualification is important. It does not invalidate the sign-degeneracy finding. It narrows its domain:

Pure cyclostrophic balance is directionally degenerate for pressure-minimum vortices, but it cannot alone explain a persistent spiral or closed circulation around a pressure maximum.

A stationary-planar model of high-pressure circulation must therefore explain two things rather than one:

Why air diverging from a pressure maximum follows a curved rather than radial trajectory.
What selects the preferred direction of that curvature.

A background rotating atmosphere, lateral body force, vertically transmitted boundary stress, or structured environmental shear could supply the required turning. Pure cyclostrophic balance cannot.

IV. Mechanisms Available on a Stationary Planar Domain

A. Background Atmospheric Circulation

A stationary surface does not logically require a stationary atmosphere. Let the background atmosphere possess an azimuthal velocity:

U_θ = U_θ(r, z, t)

Its vertical background vorticity is:

ζ_b = (1/r) [∂(rU_θ)/∂r]

A local pressure system embedded in this flow interacts with ambient vorticity and angular-momentum gradients. Radially moving air does not necessarily retain the tangential velocity of its new surroundings. The resulting relative motion can curve trajectories and bias developing vortices toward one rotational sign.

Atmospheric superrotation research establishes that an atmosphere can possess an angular-momentum distribution substantially different from simple surface corotation. Numerical and theoretical studies attribute such states to wave transport, eddy momentum fluxes, convection, and meridional circulation.⁵ These studies are not evidence for a stationary plane, but they demonstrate a relevant dynamical principle: the motion of an atmosphere need not be mechanically identical to the motion of the surface beneath it.

For a stationary-planar model, the background-flow hypothesis would require measurement of U_θ(r, z, t), identification of the torque maintaining it, and demonstration that its vorticity field predicts the directional statistics of local pressure systems.

A radial sign reversal could be represented by a velocity profile in which ζ_b changes sign at a characteristic radius. The resulting domain would contain two circulation regimes separated by a transition zone. This is mathematically capable of producing opposite directional preferences in different regions without invoking spherical hemispheres. The model’s validity would depend on whether such a background field is observed independently of the weather systems it is intended to explain.

B. Angular-Momentum and Torque Selection

Define specific angular momentum around a chosen axis as:

h = rv_θ

Its material change is governed by tangential force or torque:

Dh/Dt = rF_θ

Cyclostrophic radial balance may be indifferent to the sign of v_θ, while the angular-momentum equation is not. A positive or negative torque selects a sign.

Potential torque sources in a stationary-planar model include:

Frictional exchange with an already moving atmospheric layer.
Stress transmitted from an upper boundary.
Differential heating that drives systematic overturning and shear.
Electromagnetic or plasma interaction.
Acoustic or wave momentum deposition.
Topographic and coastal stress.
An external moving medium.

This model class can explain organized directional patterns if the torque field is specified before examining the resulting vortex directions. Defining the torque retrospectively from the observed weather would make the model descriptive but not independently predictive.

C. Boundary-Driven Circulation

Laboratory fluid experiments demonstrate that a stationary container can contain persistent vortices when one or more boundaries move. Experiments with moving sidewalls generate steady and time-dependent cellular vortex structures, while rotating endwalls can generate confined swirling flows and vortex breakdown.⁶

The atmospheric analogue would involve a stationary lower plane and an upper or lateral boundary that transfers tangential momentum to the air. A generalized vertical shear stress may be represented as:

τ_θz = μ(∂U_θ/∂z)

Here, μ represents effective viscosity or turbulent momentum transfer.

A domain-wide moving boundary could create a common circulation direction. Differential boundary motion could generate concentric, counterrotating, or otherwise structured cells. Local high-pressure outflow embedded in those cells could be curved by the environmental shear.

This mechanism is physically coherent in the abstract. For terrestrial application it requires evidence for the boundary, measurement of its motion or stress, and closure of the energy budget. The relevant question is not whether boundary-driven vortices are possible; experiments show that they are. The question is whether a boundary with the necessary properties exists and produces quantitatively correct atmospheric effects.

D. Thermal and Baroclinic Production

Nonrotating radiative-convective simulations produce organized convection, moisture contrasts, overturning circulations, and self-aggregation. Simulations comparing rotating and nonrotating configurations show that rotation changes organization, stability, humidity, precipitation, and vortex behavior, but thermodynamic organization itself does not vanish when rotation is removed.⁷

On a planar domain, a moving or spatially structured heat source could establish:

Radial or annular temperature gradients.
Repeated convergence and divergence zones.
Vertical circulation cells.
Density-pressure misalignment.
Shear layers.
Baroclinically generated vorticity.

The baroclinic term is:

(∇ρ × ∇p)/ρ²

This term can generate vorticity wherever density and pressure surfaces are not parallel. Environmental shear can then tilt or reorganize horizontal vorticity into vertical rotation.

Thermal and baroclinic mechanisms can therefore generate vortices without an initially rotating surface. They do not automatically produce a stable domain-wide preference for clockwise or counterclockwise motion. Such a preference would require asymmetry in the heating path, environmental shear, boundary geometry, or another sign-selecting influence.

A moving heat source could supply this asymmetry. Its trajectory could create a recurring wake or shear field. To become a complete explanation, however, the model would need to predict the sign, magnitude, vertical structure, seasonal movement, and persistence of observed circulation rather than merely asserting that heating produces motion.

E. A Velocity-Dependent Transverse Field

The most general stationary-planar alternative introduces a field that acts perpendicular to motion:

F_T = K(r, z, t) (z-axis unit vector × u)

This field would curve outward flow from a pressure maximum and could also select vortex direction. If K changed sign across a radial transition region, the model could produce opposite directional preferences in different portions of the plane.

Possible interpretations might include a moving material medium, electromagnetic coupling, an anisotropic property of space, or another presently unspecified interaction. The mathematical possibility is straightforward. The physical burden is substantial.

A viable field theory must provide:

A measurable source for K.
A reason the force is proportional to velocity.
A predicted radial, vertical, and temporal structure.
An energy or momentum-exchange mechanism.
Effects on matter or instruments outside atmospheric weather.
Predictions that differ from a rotating-coordinate model.

Without those elements, K is a fitting parameter. With them, it becomes a testable physical hypothesis.

F. Hybrid Models

No requirement exists that a stationary-planar theory use only one mechanism. A hybrid model might combine:

A globally moving upper atmosphere.
Thermal forcing from a moving source.
Baroclinic vorticity production.
Boundary stress.
Topographic modification.
A weaker transverse field.

Conventional atmospheric models are themselves hybrid systems containing multiple interacting processes. It would therefore be inconsistent to require a planar alternative to explain every phenomenon with one elementary force.

The proper constraint is not simplicity in the sense of having only one mechanism. It is disciplined complexity: every additional mechanism must be independently specified, measured where possible, and penalized during model comparison.

V. Existing Planar-Earth Meteorological Studies and Their Scope

Two frequently cited preprints have modeled meteorology or oceanography on a flat disk. John P. Boyd’s Meteorology and Oceanography on a Flat Earth describes a nonrotating disk and uses it as a teaching contrast with standard geophysical-fluid models. Charly de Marez and Mathieu Le Corre similarly modeled a particular flat-Earth ocean and compared the simulated circulation with ocean observations.⁸

These works are relevant but limited in scope for the present question.

First, both are arXiv preprints rather than peer-reviewed demonstrations of all possible planar models. Second, they test particular combinations of geometry, forcing, rotation assumptions, and boundary conditions. Their conclusions apply directly to those implementations. They do not logically exclude a stationary plane with a rotating atmosphere, moving upper boundary, external transverse field, radially reversing torque, or another unmodeled mechanism.

This does not mean their results are unimportant. They demonstrate that a simple nonrotating disk with otherwise familiar forcing produces circulation substantially different from the conventional model. Their results therefore impose a useful constraint:

A viable stationary-planar theory cannot merely delete surface rotation from an existing atmospheric model. It must supply replacement dynamics that quantitatively recover the observed organization.

The appropriate response is neither to treat those simulations as universal disproofs nor to disregard them. They should be incorporated as baseline null models against which more fully specified planar alternatives are tested.

VI. A Model-Neutral Empirical Program

A. Separate Raw Observations from Geometry-Embedded Products

Reanalysis products such as ERA5 combine observations with a numerical atmospheric model and a predetermined coordinate geometry. ERA5 is exceptionally useful for meteorological research, but it is not assumption-neutral evidence in a contest over the geometry embedded in its assimilation system.⁹

A neutral comparison should begin, where feasible, with station-level or instrument-level records:

Barometric pressure.
Wind speed and instrument-relative direction.
Temperature.
Humidity.
Timestamp.
Surveyed instrument orientation.
Altitude or vertical position.
Direct radar or radiosonde velocity measurements.

NOAA’s Integrated Surface Database contains global hourly and synoptic observations, including wind direction, wind speed, pressure, temperature, and related variables.¹⁰ Such data could be reconstructed separately under competing coordinate models.

Tropical cyclone archives such as IBTrACS may provide a secondary test set, although their positions and tracks are already encoded in conventional geodetic coordinates.¹¹ A rigorous study would retain the original observations and then transform them under each candidate geometry rather than treating one coordinate representation as observationally primitive.

B. Directional-Pattern Test

For each pressure system, researchers should estimate:

Pressure-center location under each geometry.
Radial pressure gradient.
Radial and tangential wind components.
Vertical wind shear.
System translation.
Local environmental vorticity.
Distance from any proposed central axis or transition ring.
Boundary-layer friction.
Thermal and topographic influences.

The response variable would not simply be “clockwise” or “counterclockwise.” It would include the full velocity field, radial divergence, tangential acceleration, vertical structure, and temporal development.

Each model must predict these quantities before the cases are revealed.

C. Torque and Energy Test

A stationary-planar model involving a moving atmosphere or boundary must identify the source of atmospheric angular momentum. Researchers should calculate:

∂H/∂t = T_surface + T_upper + T_external + T_transport

Here, H represents domain-integrated atmospheric angular momentum.

A model that predicts continuous atmospheric motion without identifying a compensating torque or energy source remains incomplete. The same accounting standard should be applied to every model, including the conventional one.

D. Local Rotation-Sensor Test

Ring-laser gyroscopes, atom-based comagnetometers, and chip-scale Sagnac gyroscopes have measured persistent local rotational signals.¹² Such instruments provide an important independent constraint, but the raw signal and its interpretation should be separated.

A discriminating experiment would deploy calibrated instruments:

At multiple positions across the proposed domain.
At different elevations and depths.
In different orientations.
In vacuum and controlled media.
With electromagnetic shielding.
On mechanically isolated platforms.
Alongside direct atmospheric-motion measurements.

A rotating-surface model predicts one spatial and orientational relationship. A stationary surface with a moving atmosphere predicts that sufficiently isolated inertial instruments should not simply follow atmospheric velocity. An external-field model must predict how the field couples to photons, atoms, mechanical gyroscopes, and atmospheric parcels. A boundary-driven atmospheric model should produce atmospheric rotation without necessarily producing the same signal in sealed inertial sensors.

The goal is not to label the signal in advance, but to determine which candidate mechanism predicts its behavior across different sensor types and environments.

E. Preregistered Model Comparison

Each model should publish in advance:

Governing equations.
Free parameters.
Initial and boundary conditions.
Expected circulation patterns.
Predicted sensor signals.
Energy and momentum budgets.
Uncertainty ranges.
Failure criteria.

Models should then be compared using withheld observations. Fit quality alone is insufficient because a flexible model can reproduce known data through parameter adjustment. Complexity penalties, out-of-sample prediction, and independent measurement of proposed forces are essential.

VII. Findings

Finding 1: Cyclostrophic balance does not determine rotational direction.

The elementary cyclostrophic equation contains v_θ². It determines the speed required for radial balance but does not select clockwise or counterclockwise flow.

Finding 2: Pure cyclostrophic balance is not a complete explanation for high-pressure circulation.

A pressure maximum drives outward acceleration. A sustained curved or spiral outflow requires another turning influence. Any stationary-planar explanation of circulation around high pressure must specify that influence.

Finding 3: Organized vorticity does not uniquely require a rotating surface.

Baroclinic torque, shear, convection, vortex stretching, moving walls, upper-boundary stress, and imposed body forces can generate or organize circulation in fluids.

Finding 4: A stationary-planar model can reproduce domain-wide directional preferences in principle.

It can do so through a background atmospheric vorticity field, spatially structured torque, moving boundary, velocity-dependent transverse force, or hybrid combination. A radial sign change can replace a hemispheric sign change mathematically.

Finding 5: Mathematical possibility is not equivalent to evidentiary support.

A model gains scientific weight by independently predicting observations, not merely by accommodating them after the fact. A planar model must identify the source and measurable properties of its sign-selecting mechanism.

Finding 6: Existing flat-disk simulations test limited model classes.

Published preprints comparing conventional circulation with a simple nonrotating disk are relevant baseline studies, but they do not evaluate every stationary-planar model. Their negative results require more sophisticated alternatives to provide replacement dynamics; they do not eliminate such alternatives by definition.

VIII. Limitations

This paper presents a theoretical framework rather than a completed empirical model. It does not derive a unique function for background atmospheric rotation, upper-boundary stress, or a transverse field. It therefore establishes constructibility and test requirements, not confirmation.

Several proposed mechanisms may prove incompatible with measured energy budgets, vertical wind profiles, gyroscope behavior, or pressure-wind relationships. Conversely, a conventional rotating-surface model should not be treated as exempt from explicit testing merely because it is widely used. Equal methodological treatment means applying common data, uncertainty analysis, and predictive standards to all candidates. It does not mean assigning equal probability to models regardless of their performance.

The word “planar” also describes a broad class rather than one model. Different planar maps, boundary shapes, source trajectories, vertical structures, and force laws produce different predictions. Peer review should therefore reject broad claims about what “a flat Earth model” necessarily predicts unless the model’s actual equations and boundaries are specified.

IX. Conclusion

Atmospheric circulation cannot be analyzed rigorously by reducing the question to whether high pressure “must” rotate in one direction. The observed flow results from a complete system of pressure gradients, inertia, divergence, shear, thermodynamics, boundaries, torques, and possibly other body forces.

Cyclostrophic balance supplies an important but limited result. It is directionally sign-degenerate for pressure-minimum vortices because tangential velocity appears squared. It does not, by itself, sustain circulation around a pressure maximum. A high-pressure system on a stationary plane can nevertheless develop curved, directionally organized outflow if embedded in a rotating atmosphere, acted upon by a moving boundary, subjected to structured environmental shear, or coupled to a transverse body field.

The central issue is therefore not whether alternative models can be imagined. They can. Nor is it whether the conventional model is merely “consensus.” It is whether each model can specify its mechanisms before examining the outcome, close its energy and angular-momentum budgets, reproduce the full three-dimensional velocity and pressure fields, and make successful predictions across independent instruments.

An assumption-neutral investigation does not prohibit conventional geometry. It removes its status as an unstated premise. It likewise does not grant a stationary-planar model immunity from evidentiary burdens. It places all proposed geometries and mechanisms within the same framework of explicit equations, measurable causes, and falsifiable predictions.

Notes

Nadir Jeevanjee, Pedram Hassanzadeh, Spencer Hill, and Aditi Sheshadri, “A Perspective on Climate Model Hierarchies,” Journal of Advances in Modeling Earth Systems 9, no. 4 (2017): 1760–71, doi:10.1002/2017MS001038; Penelope Maher et al., “Model Hierarchies for Understanding Atmospheric Circulation,” Reviews of Geophysics 57 (2019). NOAA repository record.
Robert Davies-Jones, “A Lagrangian Model for Baroclinic Genesis of Mesoscale Vortices. Part I: Theory,” Journal of the Atmospheric Sciences 57 (2000): 715–36. Davies-Jones’s formulation analyzes baroclinic vorticity production in atmospheric flow. Journal record.
Roger K. Smith, “The Cyclostrophic Adjustment of Vortices with Application to Tropical Cyclone Modification,” Journal of the Atmospheric Sciences 38 (1981): 2021–30; Vincent T. Wood and Luther W. White, “A Parametric Wind–Pressure Relationship for Rankine versus Non-Rankine Cyclostrophic Vortices,” Journal of Atmospheric and Oceanic Technology 30 (2013). Smith article record.
Hugh E. Willoughby, “The Golden Radius in Balanced Atmospheric Flows,” Monthly Weather Review 139 (2011), which explicitly notes that cyclostrophic winds are impossible in high-pressure systems; Keith F. Brill, “Revisiting an Old Concept: The Gradient Wind,” Monthly Weather Review 142 (2014): 1460–71. Willoughby article record.
Isaac M. Held and Arthur Y. Hou, “Nonlinear Axially Symmetric Circulations in a Nearly Inviscid Atmosphere,” Journal of the Atmospheric Sciences 37, no. 3 (1980): 515–33; Anne L. Laraia and Tapio Schneider, “Superrotation in Terrestrial Atmospheres,” Journal of the Atmospheric Sciences 72, no. 11 (2015): 4281–96; Neil T. Lewis, Greg J. Colyer, and Peter L. Read, “Characterizing Regimes of Atmospheric Circulation in Terms of Their Global Superrotation,” Journal of the Atmospheric Sciences 78, no. 4 (2021): 1245–58. Held and Hou record.
Christian Blohm and Hendrik C. Kuhlmann, “The Two-Sided Lid-Driven Cavity: Experiments on Stationary and Time-Dependent Flows,” Journal of Fluid Mechanics 450 (2002): 67–95; Igor V. Naumov and Irina Yu. Podolskaya, “Topology of Vortex Breakdown in Closed Polygonal Containers,” Journal of Fluid Mechanics 820 (2017): 263–83. The studies experimentally demonstrate vortex generation and organization by moving boundaries within otherwise stationary containers. Cambridge article record.
David Coppin and Romain Roehrig, “Convection Self-Aggregation in CNRM-CM6-1: Equilibrium and Transition Sensitivity to Surface Temperature,” Journal of Advances in Modeling Earth Systems 14 (2022): e2022MS003064; J. D. Carstens and A. A. Wing, “A Spectrum of Convective Self-Aggregation Based on Background Rotation,” Journal of Advances in Modeling Earth Systems 14 (2022): e2021MS002860; Levi G. Silvers, Alyssa M. Stansfield, and Kevin A. Reed, “The Impact of Rotation on Tropical Climate, the Hydrologic Cycle, and Climate Sensitivity,” Geophysical Research Letters 51 (2024): e2023GL105850. Coppin and Roehrig record.
John P. Boyd, “Meteorology and Oceanography on a Flat Earth,” arXiv preprint arXiv:2003.08541, revised March 27, 2020; Charly de Marez and Mathieu Le Corre, “Can the Earth Be Flat? A Physical Oceanographer’s Perspective,” arXiv preprint arXiv:2001.01521, January 6, 2020. Both records identify the works as arXiv submissions; Boyd describes his paper as a teaching aid and models a nonrotating disk. Boyd preprint.
Hans Hersbach et al., “The ERA5 Global Reanalysis,” Quarterly Journal of the Royal Meteorological Society 146, no. 730 (2020): 1999–2049. ERA5 is a model-based global reanalysis produced through data assimilation rather than a collection of uninterpreted raw observations. ERA5 article.
National Centers for Environmental Information, “Global Hourly: Integrated Surface Database,” documenting global hourly and synoptic station measurements including pressure, wind direction, wind speed, temperature, and humidity. Integrated Surface Database.
Kenneth R. Knapp, Michael C. Kruk, David H. Levinson, Howard J. Diamond, and Charles J. Neumann, “The International Best Track Archive for Climate Stewardship: Unifying Tropical Cyclone Data,” Bulletin of the American Meteorological Society 91, no. 3 (2010): 363–76. IBTrACS article record.
Karl Ulrich Schreiber et al., “Direct Measurement of Diurnal Polar Motion by Ring Laser Gyroscopes,” Journal of Geophysical Research 109 (2004); Rui Zhang Li et al., “Rotation Sensing Using a K-Rb-²¹Ne Comagnetometer,” Physical Review A 94 (2016): 032109; Yu-Hung Lai, Myoung-Gyun Suh, and Kerry Vahala, “Earth Rotation Measured by a Chip-Scale Ring Laser Gyroscope,” Nature Photonics 14 (2020): 345–49. These studies report local rotational measurements using distinct instrument classes. Schreiber article record.

Bibliography

Blohm, Christian, and Hendrik C. Kuhlmann. “The Two-Sided Lid-Driven Cavity: Experiments on Stationary and Time-Dependent Flows.” Journal of Fluid Mechanics 450 (2002): 67–95. doi:10.1017/S0022112001006267.
Boyd, John P. “Meteorology and Oceanography on a Flat Earth.” arXiv preprint arXiv:2003.08541. Revised March 27, 2020.
Brill, Keith F. “Revisiting an Old Concept: The Gradient Wind.” Monthly Weather Review 142 (2014): 1460–71. doi:10.1175/MWR-D-13-00088.1.
Carstens, J. D., and A. A. Wing. “A Spectrum of Convective Self-Aggregation Based on Background Rotation.” Journal of Advances in Modeling Earth Systems 14 (2022): e2021MS002860. doi:10.1029/2021MS002860.
Coppin, David, and Romain Roehrig. “Convection Self-Aggregation in CNRM-CM6-1: Equilibrium and Transition Sensitivity to Surface Temperature.” Journal of Advances in Modeling Earth Systems 14 (2022): e2022MS003064. doi:10.1029/2022MS003064.
Davies-Jones, Robert. “A Lagrangian Model for Baroclinic Genesis of Mesoscale Vortices. Part I: Theory.” Journal of the Atmospheric Sciences 57 (2000): 715–36.
de Marez, Charly, and Mathieu Le Corre. “Can the Earth Be Flat? A Physical Oceanographer’s Perspective.” arXiv preprint arXiv:2001.01521. January 6, 2020.
Held, Isaac M., and Arthur Y. Hou. “Nonlinear Axially Symmetric Circulations in a Nearly Inviscid Atmosphere.” Journal of the Atmospheric Sciences 37, no. 3 (1980): 515–33. doi:10.1175/1520-0469(1980)037<0515:NASCIA>2.0.CO;2.
Hersbach, Hans, et al. “The ERA5 Global Reanalysis.” Quarterly Journal of the Royal Meteorological Society 146, no. 730 (2020): 1999–2049. doi:10.1002/qj.3803.
Jeevanjee, Nadir, Pedram Hassanzadeh, Spencer Hill, and Aditi Sheshadri. “A Perspective on Climate Model Hierarchies.” Journal of Advances in Modeling Earth Systems 9, no. 4 (2017): 1760–71. doi:10.1002/2017MS001038.
Knapp, Kenneth R., Michael C. Kruk, David H. Levinson, Howard J. Diamond, and Charles J. Neumann. “The International Best Track Archive for Climate Stewardship: Unifying Tropical Cyclone Data.” Bulletin of the American Meteorological Society 91, no. 3 (2010): 363–76. doi:10.1175/2009BAMS2755.1.
Lai, Yu-Hung, Myoung-Gyun Suh, and Kerry Vahala. “Earth Rotation Measured by a Chip-Scale Ring Laser Gyroscope.” Nature Photonics 14 (2020): 345–49.
Laraia, Anne L., and Tapio Schneider. “Superrotation in Terrestrial Atmospheres.” Journal of the Atmospheric Sciences 72, no. 11 (2015): 4281–96. doi:10.1175/JAS-D-15-0030.1.
Lewis, Neil T., Greg J. Colyer, and Peter L. Read. “Characterizing Regimes of Atmospheric Circulation in Terms of Their Global Superrotation.” Journal of the Atmospheric Sciences 78, no. 4 (2021): 1245–58. doi:10.1175/JAS-D-20-0326.1.
Maher, Penelope, et al. “Model Hierarchies for Understanding Atmospheric Circulation.” Reviews of Geophysics 57 (2019). doi:10.1029/2018RG000607.
Naumov, Igor V., and Irina Yu. Podolskaya. “Topology of Vortex Breakdown in Closed Polygonal Containers.” Journal of Fluid Mechanics 820 (2017): 263–83. doi:10.1017/jfm.2017.211.
Silvers, Levi G., Alyssa M. Stansfield, and Kevin A. Reed. “The Impact of Rotation on Tropical Climate, the Hydrologic Cycle, and Climate Sensitivity.” Geophysical Research Letters 51 (2024): e2023GL105850. doi:10.1029/2023GL105850.
Smith, Roger K. “The Cyclostrophic Adjustment of Vortices with Application to Tropical Cyclone Modification.” Journal of the Atmospheric Sciences 38 (1981): 2021–30.
Willoughby, Hugh E. “The Golden Radius in Balanced Atmospheric Flows.” Monthly Weather Review 139 (2011).
Wood, Vincent T., and Luther W. White. “A Parametric Wind–Pressure Relationship for Rankine versus Non-Rankine Cyclostrophic Vortices.” Journal of Atmospheric and Oceanic Technology 30 (2013).

Mathematical Appendix

Candidate Models for Atmospheric Circulation without Presupposed Global Geometry

Status of this appendix: The equations below define candidate model classes for comparison. They do not, by themselves, prove a spherical surface, a planar surface, a rotating surface, or a stationary surface. A model becomes an empirical explanation only when its forcing terms, parameters, boundary conditions, and predictions are independently measured and successfully tested against withheld observations.

A. Purpose and Scope

This appendix formalizes the principal atmospheric models discussed in the accompanying essay. The objective is to make each model explicit enough that it can be simulated, fitted to observations, and rejected if its predictions fail.

The framework begins with conservation equations that do not initially prescribe the global shape or motion of the lower boundary. Geometry and surface motion enter only through the domain, coordinate definitions, force terms, and boundary conditions selected for a particular model.

The models considered are:

A radial pressure-flow null model.
A pure cyclostrophic vortex model.
A rotating-coordinate comparison model.
A stationary plane with an independently moving atmosphere.
A stationary plane with a moving upper or lateral boundary.
A thermally and baroclinically forced planar model.
A velocity-dependent transverse-field model.
A moving external-medium model.
An electromagnetic or plasma-coupled model.
A hybrid model combining multiple mechanisms.

The models are not treated as equally established. They are treated as sufficiently definable to permit direct mathematical and experimental comparison.

B. Coordinate-Neutral Conservation Equations

B.1 Variables

The following notation is used:

u = fluid velocity vector.
u_r = radial velocity.
u_θ = tangential or azimuthal velocity.
u_z = vertical velocity.
p = pressure.
ρ = mass density.
T = temperature.
g = gravitational or downward body acceleration.
μ = dynamic viscosity.
ν = kinematic viscosity.
κ_T = thermal diffusivity.
F_ext = externally imposed force per unit volume.
f_ext = externally imposed acceleration per unit mass.
ω = fluid vorticity, defined as ∇ × u.

Vectors are shown in bold. Ordinary italic letters represent scalar quantities.

B.2 Conservation of Mass

For a compressible fluid:

∂ρ/∂t + ∇ · (ρu) = 0

This equation states that mass cannot appear or disappear within the modeled atmospheric domain except through an explicitly defined source or boundary flux.

B.3 Conservation of Momentum

A general momentum equation is:

∂u/∂t + (u · ∇)u = −(1/ρ)∇p + g + (1/ρ)∇ · τ + f_ext

Here, τ is the viscous or turbulent stress tensor. The term f_ext is left unspecified until a particular model is selected.

This is the critical location at which competing interpretations enter. A rotating-coordinate model inserts rotational terms. A moving-medium model inserts drag and transverse coupling. A boundary-driven model obtains momentum from its boundary conditions. A field model inserts a separately defined body acceleration.

B.4 Thermal Evolution

A simplified temperature equation is:

∂T/∂t + u · ∇T = κ_T∇²T + Q/(ρc_p)

The quantity Q represents radiative, chemical, electrical, latent-heat, boundary, or other thermal forcing per unit volume. The specific heat at constant pressure is c_p.

B.5 Equation of State

An ideal-gas closure may be used where appropriate:

p = ρRT

More complex closures may be substituted without changing the basic model-comparison structure.

B.6 Vorticity Evolution

The vorticity equation may be written schematically as:

Dω/Dt = (ω · ∇)u − ω(∇ · u) + (∇ρ × ∇p)/ρ² + ∇ × f_ext + viscous terms

The terms represent, respectively:

vortex stretching and tilting;
changes caused by convergence or divergence;
baroclinic generation caused by misaligned pressure and density gradients;
vorticity generated by the curl of an external acceleration;
viscous production, transport, and dissipation.

Baroclinic production of atmospheric vorticity has been developed in Lagrangian atmospheric models and applied to mesoscale vortex formation.¹

C. Model N: Radial Pressure-Flow Null Model

C.1 Assumptions

The lower surface is stationary.
No global geometry is imposed beyond a locally planar or axisymmetric domain.
There is no background tangential flow.
There is no tangential body force.
There is no moving boundary that supplies angular momentum.
The initial tangential velocity is zero.

C.2 Radial Motion

For an axisymmetric pressure field, the simplified radial equation is:

Du_r/Dt = −(1/ρ)(∂p/∂r) − γ_ru_r

The coefficient γ_r represents simplified friction or drag.

For a pressure maximum at the center, pressure decreases outward:

∂p/∂r < 0

The pressure-gradient acceleration is therefore outward.

C.3 Tangential Motion

Specific angular momentum is:

h = ru_θ

Its evolution is:

Dh/Dt = rf_θ

Under the null assumptions:

f_θ = 0

and:

u_θ(t₀) = 0

Therefore:

u_θ(t) = 0

C.4 Prediction

The null model predicts radial inflow toward a pressure minimum and radial outflow from a pressure maximum. It does not generate a preferred clockwise or counterclockwise direction from initially nonrotating conditions.

C.5 Falsification Condition

The null model is inadequate wherever systematic tangential acceleration develops in the absence of measurable initial tangential momentum, asymmetric boundaries, background shear, or another force term.

D. Model C: Pure Cyclostrophic Vortex

D.1 Radial Force Balance

Take the outward radial direction as positive. For steady circular motion:

−u_θ²/r = −(1/ρ)(∂p/∂r) + f_r

Rearranging gives:

u_θ²/r = (1/ρ)(∂p/∂r) − f_r

For pure cyclostrophic balance:

f_r = 0

Thus:

u_θ²/r = (1/ρ)(∂p/∂r)

D.2 Directional Degeneracy

Where the right side is positive:

u_θ = ±√{(r/ρ)(∂p/∂r)}

The positive and negative solutions have equal speed and opposite rotational direction. The radial cyclostrophic equation does not select between them.

D.3 Pressure-Minimum Requirement

For a central pressure minimum:

∂p/∂r > 0

The pressure-gradient force points inward and can supply centripetal acceleration.

For a central pressure maximum:

∂p/∂r < 0

The pure cyclostrophic equation would require a negative squared velocity, which has no real-valued solution. Published balanced-flow analysis accordingly recognizes that pure cyclostrophic winds are unavailable in high-pressure systems.²

D.4 Generalized High-Pressure Condition

A high-pressure vortex becomes mathematically possible only if an additional inward acceleration satisfies:

(1/ρ)(∂p/∂r) − f_r > 0

Because the pressure term is negative around a pressure maximum, f_r must be sufficiently negative, meaning sufficiently inward.

D.5 Prediction

Pure cyclostrophic balance predicts the speed of a pressure-minimum vortex but not its rotational sign. It cannot independently explain persistent circulation around a pressure maximum.

D.6 Falsification Condition

The pure model fails wherever a pressure maximum supports sustained curved circulation without an independently identified inward acceleration.

E. Model R: Rotating-Coordinate Comparison Model

E.1 Purpose

This model is included as a comparator, not as a presupposition. It represents motion described in a coordinate system rotating with angular-velocity vector Ω.

E.2 Momentum Equation

Du/Dt = −(1/ρ)∇p + g + (1/ρ)∇ · τ − 2Ω × u − Ω × (Ω × r)

The two final terms are introduced by the rotating coordinate description. The first is velocity-dependent and transverse to the parcel velocity. The second depends on position relative to the selected rotation axis.

E.3 Local Turning Parameter

Under the conventional spherical implementation, a local parameter is defined as:

f = 2Ω sin(φ)

This equation is model-specific. It should not be treated as geometry-neutral evidence because φ is defined within the assumed spherical coordinate system.

E.4 Prediction

The model predicts a transverse acceleration proportional to velocity and to the locally defined parameter f. It also predicts a specific geographic and orientational pattern for inertial instruments.

E.5 Falsification Condition

The model fails if measured transverse accelerations or inertial-sensor signals systematically disagree with the spatial and directional field derived from the independently measured value of Ω.

F. Model A: Stationary Plane with Background Atmospheric Circulation

F.1 Assumptions

The lower surface is stationary.
The atmosphere possesses an independently moving background flow.
Local weather systems form within that moving atmosphere.
Direction is selected by background vorticity, angular-momentum gradients, shear, or torque.

F.2 Flow Decomposition

u = U_b + u′

The background flow is U_b. Local weather-related deviations are represented by u′.

For an axisymmetric background:

U_b = U_θ(r,z,t) in the tangential direction

F.3 Background Vorticity

ζ_b = (1/r) [∂(rU_θ)/∂r]

The sign of ζ_b identifies the local rotational tendency of the background velocity field. A local vortex can inherit, amplify, oppose, or be sheared by this background vorticity.

F.4 Smooth Example with Radial Vorticity Reversal

One example profile is:

U_θ(r) = U₀ (r/R₀) exp[−(r/R₀)²]

Its background vorticity is:

ζ_b(r) = (2U₀/R₀) exp[−(r/R₀)²] [1 − (r/R₀)²]

This field is smooth at the center. Its vorticity changes sign at:

r = R₀

The mean tangential velocity does not reverse there, but the local background vorticity does. This provides one mathematical route to two radially organized dynamical regimes on a plane.

F.5 Example with Mean-Flow Reversal

A profile whose tangential velocity itself changes direction is:

U_θ(r) = U₀ (r/R₀) tanh[({r_c − r})/L_c]

Here:

r_c is the transition radius;
L_c is the width of the transition region;
the factor r/R₀ prevents a finite tangential speed at the central point.

F.6 Parcel Angular Momentum

For a parcel with negligible applied torque:

h = ru_θ = constant

A parcel beginning at radius r₀ with the local background velocity has:

h₀ = r₀ U_θ(r₀)

After moving to radius r, its torque-free tangential velocity is:

u_θ,parcel(r) = r₀ U_θ(r₀)/r

Its velocity relative to the new environment is:

Δu_θ = [r₀ U_θ(r₀)/r] − U_θ(r)

The sign of Δu_θ determines the parcel’s tangential motion relative to its surroundings. Outward and inward radial motions can therefore acquire systematic curvature on a stationary surface when embedded in a structured moving atmosphere.

F.7 Source of Background Angular Momentum

The background atmospheric circulation must satisfy an angular-momentum budget:

∂H_a/∂t = T_lower + T_upper + T_external + T_wave + T_eddy

Studies of atmospheric superrotation demonstrate that atmospheric angular momentum can be redistributed by waves, eddies, and mean circulation rather than remaining in simple local corotation with a lower boundary.³ This establishes the dynamical possibility of substantial atmosphere-surface differential motion, although it does not establish a stationary planar surface.

F.8 Predictions

Systematic curvature should correlate with independently measured background vorticity and shear.
The directional tendency may vary with altitude if U_θ varies with z.
A radial transition should appear where the background vorticity or mean flow changes sign.
Pressure systems should drift with, or be deformed by, the background circulation.

F.9 Falsification Conditions

The model is rejected if no independently measurable background circulation or torque field exists at the magnitude needed to reproduce the observed turning.

G. Model B: Stationary Plane with a Moving Boundary

G.1 Domain

Let the atmosphere occupy:

0 ≤ z ≤ H

The lower boundary is stationary:

u_θ(r,0,t) = 0

The upper boundary moves tangentially:

u_θ(r,H,t) = U_B(r,t)

G.2 Laminar Approximation

For a simple steady shear layer with constant viscosity and negligible pressure-driven tangential flow:

u_θ(r,z) ≈ (z/H)U_B(r)

The transmitted shear stress is approximately:

τ_θz = μ(∂u_θ/∂z) ≈ μU_B/H

At atmospheric Reynolds numbers, the full turbulent equations would generally be required. The linear profile is only an explanatory limiting case.

G.3 Radially Reversing Boundary

A moving boundary with opposite directions in two radial zones can be represented as:

U_B(r) = U_B0 (r/R_B) tanh[({r_c − r})/L_B]

This boundary imposes one tangential direction inside the transition radius and the opposite direction outside it.

G.4 Experimental Basis

Moving-wall cavity experiments demonstrate that stationary containers can support organized vortices, multiple steady states, instabilities, and three-dimensional cellular structures when momentum is supplied through moving boundaries.⁴

The laboratory result establishes the fluid-mechanical possibility of boundary-driven circulation. A terrestrial application would still require independent evidence for the proposed boundary and its motion.

G.5 Predictions

Tangential velocity should generally increase toward the moving boundary.
Vertical profiles should reveal stress transmission from the boundary.
Changes in boundary speed should precede or accompany changes in atmospheric circulation.
The atmospheric energy budget should contain power supplied by the moving boundary.

G.6 Power Requirement

The rate of mechanical work per unit area supplied by the upper boundary is:

P_B = τ_θz U_B

A complete model must show that the integrated boundary power is sufficient to maintain the proposed circulation.

G.7 Falsification Conditions

The model fails if the proposed boundary cannot be detected, if its measured stress is too small, or if circulation does not exhibit the predicted vertical dependence.

H. Model T: Thermally and Baroclinically Forced Planar Circulation

H.1 Thermal Forcing

The temperature field follows:

∂T/∂t + u · ∇T = κ_T∇²T + Q/(ρc_p)

For a simplified buoyancy relation:

ρ ≈ ρ₀ [1 − α(T − T₀)]

The coefficient α is the thermal expansion coefficient.

H.2 Baroclinic Vorticity Source

B_baro = (∇ρ × ∇p)/ρ²

If pressure and density gradients are not parallel, this term can generate vorticity without requiring initial rotation of the lower surface.

H.3 Symmetry Limitation

A perfectly axisymmetric, stationary heat source does not independently distinguish clockwise from counterclockwise motion. A directional preference requires broken symmetry through:

a moving source;
asymmetric source geometry;
background shear;
unequal boundary conditions;
preexisting vorticity;
an external torque or field.

H.4 Moving Heat-Source Model

Let a localized heat source travel around a circular path on a plane:

x_s(t) = R_s cos(ω_st)

y_s(t) = R_s sin(ω_st)

A three-dimensional Gaussian heating distribution is:

Q(x,y,z,t) = Q₀ exp{−[(x − x_s)² + (y − y_s)²]/σ_h²} exp{−(z − z_s)²/σ_z²}

The sign of ω_s selects the direction in which the heat source travels. The moving temperature and pressure field may generate a directional wake, asymmetric convergence zones, and baroclinic vorticity.

H.5 Predictions

Vorticity production should correlate with the measured baroclinic term.
Circulation should exhibit a phase relationship with the moving heat source.
Reversing the source direction should reverse the associated directional bias, after accounting for initial conditions.
Removing source asymmetry should reduce or eliminate systematic handedness unless another sign selector remains.

H.6 Falsification Conditions

The model fails if the measured heating, density, and pressure fields cannot supply the observed vorticity magnitude or directional organization.

I. Model K: Velocity-Dependent Transverse Field

I.1 General Form

Introduce an acceleration perpendicular to horizontal velocity:

a_K = K(r,z,t) [z-axis unit vector × u]

In Cartesian components:

a_x = −Ku_y

a_y = Ku_x

The field coefficient K has units of inverse time.

I.2 Radial Flow

For purely radial motion:

a_θ = Ku_r

Thus:

outward flow from a pressure maximum has u_r > 0;
inward flow toward a pressure minimum has u_r < 0;
the sign of K determines the direction of curvature at each location.

I.3 Radial Sign-Reversal Field

A smooth two-zone field is:

K(r) = K₀ tanh[({r_c − r})/L_c]

This produces:

K > 0 inside the transition region;
K = 0 at r = r_c;
K < 0 outside the transition region.

A stationary planar domain can therefore possess two opposite circulation tendencies without invoking spherical hemispheres. The two regions are defined by radial position relative to the field’s transition ring.

I.4 Work and Energy

The field is perpendicular to velocity:

u · a_K = 0

Therefore, the idealized field changes direction but does no instantaneous mechanical work on the parcel. Pressure gradients, buoyancy, boundary work, or another source must still provide the parcel’s kinetic energy.

I.5 Physical Interpretations

The equation alone does not identify the field’s origin. It could represent:

a rotating-coordinate term;
coupling to a moving external medium;
an electromagnetic interaction;
an anisotropic property of the environment;
an unknown velocity-dependent body force;
a phenomenological approximation to unresolved dynamics.

The physical explanation must be tested separately from the mathematical fit.

I.6 Direct Predictions

At a fixed position, a_θ/u_r should equal K.
The turning acceleration should be linear in velocity within the applicable regime.
The directional effect should vanish at the transition radius.
The sign should reverse across the transition radius.
The ideal transverse component should not directly change parcel speed.

I.7 Falsification Conditions

The model fails if the measured transverse acceleration is not proportional to velocity, if no predicted transition is found, or if no independently measurable source of K can be identified.

J. Model M: Moving External Medium

J.1 General Coupling

Let an external medium possess velocity U_m. Define relative velocity:

v_rel = u − U_m

A combined drag and transverse coupling is:

a_m = −γ_m v_rel + κ_m [z-axis unit vector × v_rel]

The coefficient γ_m controls relaxation toward the medium velocity. The coefficient κ_m controls transverse turning.

J.2 Energy Transfer

Relative kinetic energy changes through the drag component:

v_rel · [−γ_m v_rel] = −γ_m |v_rel|²

The transverse component contributes no direct work:

v_rel · {κ_m [z-axis unit vector × v_rel]} = 0

J.3 Circulating Medium

An azimuthally moving medium may be represented as:

U_m = U_mθ(r,z,t) in the tangential direction

A sign-reversing profile can use the same hyperbolic-tangent form introduced for the moving atmosphere or boundary.

J.4 Interpretation

An ether-like proposal would be one possible interpretation of this model, but the equation does not establish an ether. It defines the consequences of coupling to any moving medium with specified velocity and interaction coefficients.

J.5 Predictions

Air velocity should relax toward the measured medium velocity.
Dissipation should occur at a rate related to γ_m.
Turning should depend on relative velocity, not absolute instrument velocity alone.
Independent detectors should measure the medium or its coupling.

J.6 Falsification Conditions

The model fails if the proposed medium is undetectable at the required coupling strength, or if measured air trajectories do not exhibit the predicted relaxation and transverse acceleration.

K. Model E: Electromagnetic or Plasma-Coupled Atmosphere

K.1 Two-Component Fluid

Let the atmosphere contain a charged component and a neutral component. The charged-fluid momentum equation is:

ρ_i Du_i/Dt = −∇p_i + q_in_i (E + u_i × B) − ρ_iν_in (u_i − u_n) + other forces

The neutral-fluid equation contains the opposite collisional momentum transfer:

ρ_n Du_n/Dt = −∇p_n + ρ_iν_in (u_i − u_n) + other forces

Here:

q_i is charged-particle charge;
n_i is charged-particle number density;
E is the electric field;
B is the magnetic field;
ν_in is the ion-neutral collision frequency.

K.2 Direction Selection

If B has a vertical component and the charged fluid moves radially, the term:

u_i × B

has a tangential component. Its sign depends on charge, magnetic-field direction, and radial-flow direction. Neutral air can acquire momentum through collisions with the charged component.

K.3 Requirements

A viable electromagnetic atmospheric model must specify and measure:

charge density;
conductivity;
electric-field strength;
magnetic-field strength and orientation;
collision rates;
vertical and geographic dependence;
energy supplied by the electromagnetic source.

K.4 Predictions

Turning should vary with ionization and conductivity.
Effects should change with electric and magnetic-field orientation.
Controlled shielding or field modification should alter the response where technically possible.
Charged and neutral components may exhibit measurable velocity differences before collisional coupling equalizes them.

K.5 Falsification Conditions

The model fails if measured charge densities, fields, and collision rates are insufficient by orders of magnitude to generate the observed momentum transfer.

L. Model H: Hybrid Atmospheric Model

L.1 Combined Momentum Equation

A hybrid model may include several mechanisms:

Du/Dt = −(1/ρ)∇p + g + (1/ρ)∇ · τ + a_background + a_boundary + a_thermal + a_K + a_medium + a_EM

L.2 Avoiding Double Counting

Terms must not be counted twice. For example:

a measured background wind should not also be inserted as an unexplained transverse field;
boundary stress already contained in the stress tensor should not be added again as a body force;
electromagnetic momentum transferred through ion-neutral collisions should not be separately added as generic drag;
coordinate-generated terms should not be added to an inertial-frame model unless the coordinate transformation requires them.

L.3 Model Discipline

Each added mechanism increases flexibility. A hybrid model must therefore report:

the number of fitted parameters;
which parameters were measured independently;
which parameters were inferred from the same weather data being explained;
parameter uncertainty;
correlation and degeneracy among parameters;
out-of-sample predictive performance.

L.4 Falsification Condition

A hybrid model is not protected from rejection by adding more terms. It fails if it cannot outperform simpler models on withheld data after accounting for its additional complexity.

M. Dimensionless Comparison Parameters

Let:

L = characteristic horizontal length;
H = characteristic vertical depth;
U = characteristic velocity;
t_a = L/U, the advective timescale.

M.1 Reynolds Number

Re = UL/ν

This compares inertial transport with viscous diffusion.

M.2 Thermal Péclet Number

Pe = UL/κ_T

This compares thermal advection with thermal diffusion.

M.3 Thermal-Forcing Parameter

G_T = gαΔTL/U²

This measures the strength of buoyancy associated with a characteristic temperature difference.

M.4 Transverse-Field Number

C_K = KL/U

This compares the transverse turning timescale with the advective timescale.

M.5 Background-Shear Number

S_b = (L/U) |∂U_θ/∂r|

M.6 Boundary-Forcing Ratio

B_U = U_B/U

M.7 Medium-Drag Number

D_m = γ_mL/U

M.8 Medium-Turning Number

C_m = κ_mL/U

M.9 Electromagnetic Turning Number

C_EM = (|q|B/m) (L/U)

For a multicomponent fluid, the effective value must include ionization, collision frequency, and neutral coupling.

N. Direct Comparison for High-Pressure Outflow

Consider air moving outward from a pressure maximum, so that:

u_r > 0

Model	Source of Tangential Motion	Direction Selector	Can It Curve High-Pressure Outflow?	Primary Independent Test
Radial null model	None	None	No, not from zero tangential initial conditions	Measure systematic tangential acceleration
Pure cyclostrophic model	Inward pressure-gradient force	Initial conditions or external torque	No, not around a pressure maximum without another inward force	Evaluate the full radial force balance
Rotating-coordinate model	Velocity-dependent coordinate term	Angular-velocity vector and location	Yes	Compare wind and inertial-sensor fields with predicted spatial dependence
Moving-atmosphere model	Background vorticity, shear, and angular-momentum gradients	Measured U_θ and ζ_b	Yes	Measure vertical and radial background wind profiles
Moving-boundary model	Boundary stress	Boundary direction and speed	Yes	Detect the boundary and measure its stress and power
Thermal-baroclinic model	Density-pressure misalignment and moving thermal asymmetry	Source trajectory, shear, or initial vorticity	Yes, when symmetry is broken	Measure the baroclinic term and source phase
Transverse-field model	a_θ = Ku_r	Sign of K	Yes	Test velocity proportionality and predicted sign-reversal location
Moving-medium model	Drag and transverse coupling to a medium	Medium velocity and coupling coefficients	Yes	Detect the medium and measure relaxation toward its velocity
Electromagnetic model	Electric, magnetic, and collisional momentum transfer	Charge and field orientation	Potentially	Measure fields, charge density, conductivity, and momentum-transfer rate

O. Required Initial and Boundary Conditions

No model is complete until its initial and boundary conditions are stated.

O.1 Initial Fields

At time t = 0, specify:

ρ(x,y,z,0) = ρ₀(x,y,z)

u(x,y,z,0) = u₀(x,y,z)

T(x,y,z,0) = T₀(x,y,z)

p(x,y,z,0) = p₀(x,y,z)

O.2 Lower Boundary

Specify whether the lower boundary is:

no slip;
partial slip;
rough and turbulent;
porous;
thermally fixed;
thermally forced;
electrically conducting;
mechanically moving.

O.3 Upper Boundary

Specify whether the upper boundary is:

open;
radiative;
stress free;
fixed;
moving;
electromagnetically active;
a material boundary;
a gradual transition rather than a discrete surface.

O.4 Lateral Boundary

A planar model must specify whether the lateral boundary is:

periodic;
reflective;
open;
solid;
moving;
thermally forced;
located at a finite radius;
approached asymptotically.

Changing these conditions can change the circulation. Conclusions about “a planar model” are incomplete unless these choices are stated.

P. Parameter Estimation and Identifiability

P.1 Observation Vector

Let the measured data be:

y_i = { p, ρ, T, u_x, u_y, u_z, electric field, magnetic field, boundary motion, sensor rotation }_i

P.2 Model Prediction

For model M with parameters θ_M:

y_i,predicted = F_M (x_i, t_i; θ_M)

P.3 Weighted Error Function

J_M = Σ_i [ y_i − y_i,predicted ]^T W_i [ y_i − y_i,predicted ]

The matrix W_i weights measurements according to uncertainty and covariance.

P.4 Identifiability Problem

Two model terms may produce similar trajectories. For example:

a_K = K [z-axis unit vector × u]

may be mathematically indistinguishable over a limited data set from a rotating-coordinate term of the same functional form. Trajectory fitting alone cannot determine physical origin.

Independent observations are therefore required. These may include:

direct boundary-motion measurements;
inertial-sensor measurements in sealed environments;
electromagnetic-field measurements;
vertical wind profiles;
energy-transfer measurements;
experiments that alter one proposed mechanism while holding others fixed.

Q. Preregistered Testing Procedure

Publish each model’s equations before examining the validation data.
Identify all fitted and independently measured parameters.
Specify the coordinate transformation used for each model.
Use the same raw observational records wherever possible.
Reserve a substantial portion of observations for blind validation.
Test pressure minima and pressure maxima separately.
Test different spatial scales separately.
Test vertical structure rather than surface winds alone.
Calculate mass, energy, and angular-momentum residuals.
Penalize unnecessary parameters and mechanisms.
Publish failures and anomalous cases rather than removing them without a preregistered rule.

R. Minimum Empirical Standards by Model

Model	Minimum Evidence Required	Evidence That Would Weaken It
Rotating-coordinate model	Consistent spatial field across atmospheric motion and independent inertial instruments	Persistent, reproducible deviations not attributable to measurement error or omitted known forces
Moving atmosphere	Measured global background flow and complete torque budget	Required flow absent or energetically unsustainable
Moving boundary	Detected boundary, measured velocity, stress, and power transfer	No boundary or insufficient stress
Thermal-baroclinic model	Measured temperature, density, and pressure gradients that reproduce observed vorticity	Wrong sign, magnitude, timing, or vertical structure
Transverse field	Independent measurement of K and successful prediction of sign reversals	No velocity proportionality or no independently detectable field
Moving medium	Direct detection of the medium and its coupling	No medium signal or incorrect relaxation behavior
Electromagnetic model	Fields and charge densities sufficient to close the force budget	Force estimates far below observed acceleration

S. Principal Mathematical Conclusions

Conclusion 1: A pressure gradient can generate radial motion without selecting a clockwise or counterclockwise direction.

Conclusion 2: Pure cyclostrophic balance contains a positive and negative tangential-velocity solution, but only where the radial pressure-gradient force points inward.

Conclusion 3: Pure cyclostrophic balance does not support a steady high-pressure vortex without an additional inward acceleration.

Conclusion 4: A stationary planar domain can mathematically support opposite regional circulation tendencies through radial changes in background vorticity, boundary motion, torque, medium motion, or a transverse-field coefficient.

Conclusion 5: The existence of a mathematical force term does not establish its physical source.

Conclusion 6: Models sharing the same mathematical form must be distinguished through independent experiments, energy budgets, boundary measurements, and sensor responses.

Conclusion 7: No global geometry should be inferred from circulation direction alone unless competing geometries and force mechanisms have first been quantitatively excluded.

T. Notes

Robert Davies-Jones, “A Lagrangian Model for Baroclinic Genesis of Mesoscale Vortices. Part I: Theory,” Journal of the Atmospheric Sciences 57 (2000): 715–36, doi:10.1175/1520-0469(2000)057<0715:ALMFBG>2.0.CO;2. Journal record.
Hugh E. Willoughby, “The Golden Radius in Balanced Atmospheric Flows,” Monthly Weather Review 139 (2011): 1164–68, doi:10.1175/2010MWR3579.1. Willoughby specifically discusses the unavailability of cyclostrophic wind solutions in high-pressure systems. Journal record.
Jonathan L. Mitchell and Geoffrey K. Vallis, “The Transition to Superrotation in Terrestrial Atmospheres,” Journal of Geophysical Research: Planets 115 (2010): E12008, doi:10.1029/2010JE003587; Sébastien Lebonnois et al., “Angular Momentum Budget in General Circulation Models of Superrotating Atmospheres,” Journal of Geophysical Research: Planets 117 (2012), doi:10.1029/2012JE004223. Mitchell and Vallis record.
Hendrik C. Kuhlmann, M. Wanschura, and H. J. Rath, “Flow in Two-Sided Lid-Driven Cavities: Non-Uniqueness, Instabilities, and Cellular Structures,” Journal of Fluid Mechanics 336 (1997): 267–99, doi:10.1017/S0022112096004727; Francesco Romanò, Stefan Albensoeder, and Hendrik C. Kuhlmann, “Topology of Three-Dimensional Steady Cellular Flow in a Two-Sided Anti-Parallel Lid-Driven Cavity,” Journal of Fluid Mechanics 826 (2017): 302–34, doi:10.1017/jfm.2017.422. Kuhlmann, Wanschura, and Rath record.

U. Bibliography

Davies-Jones, Robert. “A Lagrangian Model for Baroclinic Genesis of Mesoscale Vortices. Part I: Theory.” Journal of the Atmospheric Sciences 57 (2000): 715–36. doi:10.1175/1520-0469(2000)057<0715:ALMFBG>2.0.CO;2.
Kuhlmann, Hendrik C., M. Wanschura, and H. J. Rath. “Flow in Two-Sided Lid-Driven Cavities: Non-Uniqueness, Instabilities, and Cellular Structures.” Journal of Fluid Mechanics 336 (1997): 267–99. doi:10.1017/S0022112096004727.
Lebonnois, Sébastien, et al. “Angular Momentum Budget in General Circulation Models of Superrotating Atmospheres.” Journal of Geophysical Research: Planets 117 (2012). doi:10.1029/2012JE004223.
Mitchell, Jonathan L., and Geoffrey K. Vallis. “The Transition to Superrotation in Terrestrial Atmospheres.” Journal of Geophysical Research: Planets 115 (2010): E12008. doi:10.1029/2010JE003587.
Romanò, Francesco, Stefan Albensoeder, and Hendrik C. Kuhlmann. “Topology of Three-Dimensional Steady Cellular Flow in a Two-Sided Anti-Parallel Lid-Driven Cavity.” Journal of Fluid Mechanics 826 (2017): 302–34. doi:10.1017/jfm.2017.422.
Willoughby, Hugh E. “The Golden Radius in Balanced Atmospheric Flows.” Monthly Weather Review 139 (2011): 1164–68. doi:10.1175/2010MWR3579.1.