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.

How would you explain it like I'm…

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.

Structural Signature

Sig role-phrases:

• erodible dry sand bed — supplies grains available for saltation It is essential. Counterfactual: A rigid or cohesive surface cannot satisfy the same transport relation.
• friction velocity — represents wind shear stress acting at the surface It is essential. Counterfactual: Free-stream wind speed cannot be substituted without a profile relation.
• saltation threshold — marks onset and sustained motion before flux scaling is applicable It is essential. Counterfactual: The cubic expression should not predict continuous flux below threshold.
• grain size and sorting — modify transport efficiency through the reference ratio and empirical constant It is essential. Counterfactual: Using one constant across different sediments hides a constitutive parameter.
• air density and gravity — set fluid forcing and grain-weight scales It is essential. Counterfactual: Planetary or altitude transfer requires these quantities to change.
• mass flux per unit width — defines the formula's output and measurement frame It is essential. Counterfactual: A concentration or total catch without width and time units is not q.

What It Is Not

• 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.
• Closest near-miss. A later threshold-corrected Bagnold-type formula is a related model variant, not automatically identical to the simplest cubic 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.

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

1. Confirm aeolian saltation is the dominant transport mode.
2. Measure bed moisture, cohesion, grain size, and sorting.
3. Estimate surface shear and friction velocity under a stated profile model.
4. Determine activation or maintenance threshold for the bed.
5. Select the exact Bagnold or modified expression.
6. Compute unit-width mass flux with consistent dimensions.
7. 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.

Examples

Applied / In Practice

A steady wind over well-sorted dry sand has measured friction velocity above threshold, and unit-width mass flux is estimated from the cubic scaling.

Mapped back: regime → Dry steady saltation; driver → Surface friction velocity; output → Mass per width per time.

Applied / In Practice

Air density and gravity are changed explicitly when examining saltation under another atmosphere.

Mapped back: parameters → The dimensional factors travel; empirical constants require validation..

Applied / In Practice

Moist coastal sand is assigned the dry formula with no cohesion or threshold correction.

Mapped back: boundary → Moisture violates the stated regime..

Structural Tensions

T1 — Benchmark Simplicity versus Site Heterogeneity. A compact cubic law enables comparison while moisture, sorting, bedforms, gusts, and fetch alter real flux.

Diagnostic: Treat residuals as regime diagnostics and choose validated modifications explicitly.

T2 — Wind Speed versus Surface Shear. Routine observations report wind at height, while the formula's driver is friction velocity at the bed.

Diagnostic: Document the wind-profile conversion and roughness assumptions.

Structural–Framed Character

Dimensional variables and cubic scaling are structural within the model; empirical coefficients and applicability are environment-framed. Benchmark status does not erase the dry, steady, above-threshold assumptions.

Structural Core vs. Domain Accent

The skeleton is transport flux scaling nonlinearly with forcing. Aeolian geomorphology supplies saltation, friction velocity, threshold, sand sorting, air density, gravity, and dry-bed regime. Those commitments define the Bagnold relation.

Instantiates / Related Primes

This entry under conditions is a kind of Mathematical Relation.

• Approved root. The frozen DAG leaves the formula unparented; generic scaling-law or transport nodes do not by themselves encode its aeolian regime and dimensional factors.

• Related — saltation threshold, Bagnold number, and aeolian transport. They provide its onset condition, related dimensionless comparison, and process family.

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

• Thermodynamic System — 0.88
• Open-Channel Flow — 0.87
• Thermogravitational Cycle — 0.87
• Pouillet Effect — 0.86
• Seismic Site Effects — 0.86

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

Not to Be Confused With

• Bagnold number. Tell: A dimensionless collisional-flow ratio, not this mass-flux formula.
• Bed-load formula. Tell: Usually concerns water-driven grains under different fluid and threshold conditions.
• Wind erosion. Tell: Is the broader process and includes suspension and surface creep.
• Cubic wind-speed law. Tell: Is incomplete unless the velocity, threshold, and environmental factors are specified.

References

• Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Bagnold_formula (revision 1186023262).
• Preserved source candidate: https://data.mendeley.com/datasets/675gwk5jp7/1

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.