Bagnold formula¶

An aeolian-transport law making dry steady sand mass flux scale approximately with the cube of above-threshold friction velocity, adjusted for air, gravity, and grain size.

Version: v1 · 2026-09-28 · History
Domain-specific #: 8103
Domain group: Natural Sciences
Origin domain: Geology & Earth Sciences
Subdomains: Aeolian Geomorphology, Sediment Transport → Geology & Earth Sciences

Core Idea¶

The Bagnold formula is a benchmark scaling law for wind-driven sand saltation. In its simplest form, unit-width mass flux is proportional to air density divided by gravity, multiplied by a grain-size correction, a sorting-dependent coefficient, and the cube of friction velocity.

Its cubic dependence is meaningful only within a regime: the surface must provide dry loose grains, flow must sustain saltation above threshold, and friction velocity must represent surface shear. Moisture, cohesion, mixed sizes, gustiness, finite fetch, and alternative transport modes motivate modified formulas rather than silent reuse of the dry steady expression.

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The Windy Sand Hop Rule

When wind blows over dry sand, it makes the grains hop along the ground. The Bagnold formula says that if the wind scrapes the ground a bit harder, much, much more sand moves: twice as hard means about eight times as much sand. It only works for dry, loose sand and wind strong enough to keep grains hopping.

Sand-Blowing Cube Rule

When strong wind blows over a dry, sandy surface, sand grains get lifted and bounce along in little hops, which scientists call saltation. The Bagnold formula estimates how much sand moves past each bit of ground width. The key part is how hard the wind drags on the surface, called friction velocity. Sand flow grows with the cube of that value, so doubling it makes about eight times as much sand move. The formula also includes the air's density, gravity, the grain size and how evenly sized the grains are. It only works when the sand is dry and loose and the wind is strong enough to keep grains hopping; wet, sticky or mixed sand needs changed formulas.

Saltation Flux Scaling Law

The Bagnold formula is a benchmark scaling law for how much sand wind moves by saltation (grains bouncing along the surface). In its simplest form, the mass of sand moved per unit width is proportional to air density divided by gravity, times a correction for grain size, times a coefficient depending on how well sorted the grains are, times the cube of the friction velocity (a measure of the wind's shear on the surface). The cubic dependence means sand transport rises very steeply as the wind strengthens. It only applies in a specific regime: dry, loose grains, flow strong enough to keep saltation going above the threshold, and a friction velocity that actually represents surface shear. Moisture, cohesion, mixed grain sizes, gusts, limited fetch or other transport modes call for modified formulas.

The Bagnold formula is a benchmark scaling law for aeolian sand transport by saltation. In its simplest form, the unit-width mass flux q is proportional to (ρ/g) multiplied by a grain-size correction, a sorting-dependent empirical coefficient, and u*³, where ρ is air density, g gravitational acceleration and u* the friction (shear) velocity. The cubic dependence on u* is the formula's signature and makes flux highly sensitive to wind shear. It is meaningful only within a regime: the surface must supply dry, loose grains, the flow must sustain saltation above the transport threshold, and u* must represent the actual surface shear. Moisture, cohesion, mixed grain sizes, gustiness, finite fetch and alternative transport modes violate these assumptions. In those conditions one should use modified formulas rather than silently reusing the dry, steady expression.

Scope of Application¶

Desert geomorphology. Dry dune migration and aeolian flux are estimated.
Wind-tunnel research. Controlled beds test cubic scaling and coefficients.
Field sediment monitoring. Measured shear and traps are compared with the benchmark.
Planetary aeolian studies. Density, gravity, grains, and thresholds are rescaled with validation.

Clarity¶

Report equation variant, q units and width convention, friction-velocity derivation, threshold treatment, air density, gravity, grain-size distribution, reference diameter, sorting constant, moisture, bed condition, steadiness, and uncertainty. Do not call all descendants simply the Bagnold formula without displaying the actual expression. Inclusion test: The formula applies when a dry loose bed undergoes steady above-threshold aeolian saltation and all variables use the stated flux, shear, grain, air, and gravity conventions. Exclusion test: Wet, cohesive, submerged, or suspension-dominated transport is excluded from the simple law. Nearest boundary: A later threshold-corrected Bagnold-type formula is a related model variant, not automatically identical to the simplest cubic expression. Exit condition: The identity exits below sustained-saltation threshold or when moisture, cohesion, unsteadiness, mixed grain populations, or another transport mode dominates without a calibrated modification. Common misclassifications: It is not a formula for every sediment-transport process. It is not driven directly by arbitrary reported wind speed without conversion to friction velocity. It is not valid below the saltation threshold. It is not moisture-aware in its simplest dry-desert form. Nearest named distinctions: Bagnold number: A dimensionless collisional-flow ratio, not this mass-flux formula. Bed-load formula: Usually concerns water-driven grains under different fluid and threshold conditions. Wind erosion: Is the broader process and includes suspension and surface creep. Cubic wind-speed law: Is incomplete unless the velocity, threshold, and environmental factors are specified.

Manages Complexity¶

The formula compresses coupled airflow, grain impact, momentum transfer, and hopping trajectories into a few measurable scales. It supports first-order comparison but hides threshold hysteresis, bed evolution, vertical flux structure, intermittency, and site-specific calibration.

Abstract Reasoning¶

Confirm aeolian saltation is the dominant transport mode.
Measure bed moisture, cohesion, grain size, and sorting.
Estimate surface shear and friction velocity under a stated profile model.
Determine activation or maintenance threshold for the bed.
Select the exact Bagnold or modified expression.
Compute unit-width mass flux with consistent dimensions.
Compare against observations and bound departures caused by regime violations.

Knowledge Transfer¶

Dimensional scaling transfers among dry saltating beds when air density, gravity, grain size, threshold, and coefficient are re-established. It stops at wet cohesive beds, water transport, or suspension without a new model. The cargo is above-threshold shear-driven saltation flux, not generic wind erosion.

Relationships to Other Abstractions¶

Current abstraction Bagnold formula Domain-specific

Parents (1) — more general patterns this builds on

Bagnold formula is a kind of, conditional Mathematical Relation Domain-specific

Supported as an approximate physical mathematical relation only within its dry steady transport regime and calibration conditions.

Condition / exception Supported as an approximate physical mathematical relation only within its dry steady transport regime and calibration conditions.

Hierarchy path (1) — routes to 1 parentless root

Bagnold formula → Mathematical Relation

Neighborhood in Abstraction Space¶

Bagnold formula sits in a moderately populated region (43^rd percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Thermodynamic & Transport Processes (34 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08