How Wildfires Learn to Jump: The Science of Ember Transport

The Invisible Army That Lights New Fires

Some fires don't spread the way you'd expect. They don't creep steadily across the landscape, consuming fuel in a neat, predictable wave. Instead, they leap. They skip. They rain embers across hillsides and valleys, igniting spot fires miles ahead of the main flame front, catching communities off-guard, turning battles against wildfire into something closer to defense against chaos.

Consider this: in a single large wildfire, updrafts can exceed 100 miles per hour—strong enough to loft burning branches high into the atmosphere, where winds carry them over the horizon. These embers, or "firebrands" in the fire-science community, can land tens of miles away, starting new fires long after the original front has passed. In the 2018 Camp Fire, which killed 85 people and destroyed nearly 19,000 structures in Paradise, California, investigators found evidence of spot fires igniting far ahead of the main fire line—fires that residents had no warning to escape. The inferno didn't just march toward them; it materialized around them.

This is the story of how fires spread in ways that defy simple explanation, and how researchers are finally building the mathematical language to describe it. A new chapter by Kevin Speer, Bryan Quaife, and Jie Sun—published through the arXiv preprint server—assembles decades of fire science into something remarkable: a coherent framework for understanding ember transport, from the physics of a burning twig to the statistical patterns of where thousands of embers might land.

The work matters because it addresses one of the most frightening aspects of wildfire: unpredictability. When flames can outrun human evacuation routes, when fires "spot" ahead of themselves, when a calm morning can transform into an apocalyptic afternoon, science needs to do better than anecdote. This paper is a step toward that science—a way of seeing fire not as a monolithic disaster but as a cascade of individual embers, each one following physical laws that, while complex, are now being written down in equations humans can study, test, and eventually predict.

The Science of Moving Fire

Wildland fires are what scientists call "multiscale phenomena"—events whose behavior emerges from interactions happening at tiny scales (the chemistry of combustion, the turbulence around a single flame) and enormous scales (atmospheric boundary layers, weather systems, climate trends). The spread of fire depends on radiant and convective heat fluxes, combustion chemistry, the type and arrangement of fuel, terrain, and atmospheric conditions. At the extreme end, conditions with winds reaching hurricane strength may be the progenitor of extreme wildfires—or may be caused by them, in a feedback loop where fire releases enough energy to modify the larger-scale atmosphere, sometimes producing clouds, precipitation, and violent downdrafts and updrafts.

The authors' approach is to build what they call a "hierarchy of modeling approaches," ranging from simple idealized models to full-physics atmospheric simulations. This hierarchy matters because different fire scenarios require different levels of complexity. Emergency managers need rapid, operational models that can run in real time during an active incident. Researchers need more detailed simulations to understand the fundamental processes. The authors aim to present "the minimal models capable of reproducing key facets of observed fires."

Their framework views ember spotting as a chain of five factors:

$$\boxed{\text{Fire Area}}\times \boxed{\text{Ember Source}}\times \boxed{\text{Ember Transport}}\times \boxed{\text{Landing Distribution}}\times \boxed{\text{Probability of Ignition}}$$

Each factor is its own field of study. Fire area determines how many embers can potentially be generated. The ember source term represents how many burning pieces are released from vegetation or structures under given conditions. Ember transport describes how those pieces move through the air, subject to drag, gravity, and turbulent wind. The landing distribution tells you where embers are likely to fall. And the probability of ignition—dependent on temperature, humidity, time of day, fuel type, and shading—determines whether a landing ember actually starts a new fire.

To understand how embers behave, the researchers draw on three types of evidence: controlled laboratory settings, where variables can be isolated and measured precisely; prescribed fires, which are intentionally set by land managers under controlled conditions; and observations from actual wildland fires, where conditions are messier but the stakes are real. Satellite data—particularly newer infrared products—has begun providing better coverage of fire perimeters and spot fires, though distinguishing genuine spot ignitions from artifacts in the data remains challenging.

What Makes Fire Spread

Before understanding how embers travel, it's worth understanding how fire itself moves. The "rate of spread" (RoS) measures how fast the fire front advances, typically expressed as velocity normal to the flame front. A comprehensive synthesis of observations suggests that the downwind front spread rate in forests and shrublands is about 10% of wind speed—with a much larger but less well-determined rate in grasslands. Strong winds can easily drive fires faster than people can run.

The RoS is not a single number. Research using outdoor laboratory settings with imposed wind found that the forward component of RoS follows an exponential distribution, roughly consistent with relatively laminar or weakly turbulent fixed wind conditions. But when researchers analyzed high-resolution data from prescribed fires conducted outdoors under realistic atmospheric conditions, a more complex picture emerged. The wind field in these natural settings is stronger and far more turbulent, including the effects of the atmospheric boundary layer—the lowest kilometer or so of the atmosphere, where wind speed increases with height and turbulence shapes the flow.

In these realistic conditions, the distribution of spread rates splits into two regimes. The lower spread rates follow a Pareto distribution—a mathematical pattern where most values cluster near a characteristic scale (in this case, an expected value of about 0.5 meters per second, given a scale parameter of 0.25 m/s and shape parameter of 2), but the distribution has a heavy tail containing rare but extreme values. Some fires in the dataset spread at rates exceeding 10 meters per second—roughly 22 miles per hour of forward advance. This suggests a lack of a characteristic spread rate scale and the presence of aggregation or multiplicative effects, possibly better described as a cascade process where small variations amplify into extreme events.

The figure above shows this distribution from data collected by Paugam et al. The Pareto distribution describes the lower spread rates, while the tail reveals the rare, dangerous outliers. Understanding this tail matters enormously for fire management: these are the fires that overwhelm containment lines, the fires that create the conditions for catastrophic spotting.

At larger scales, fires can grow in two fundamentally different ways. Some fires exhibit linear area growth—the burned area scales with time ($A(t)\sim t$). Others exhibit quadratic growth—the area scales with time squared ($A(t)\sim t^2$). These aren't just mathematical curiosities; they reflect different physical behaviors. Linear growth tends to occur in fires with minimal spotting, where the fire spreads primarily from the main front. Quadratic growth tends to occur in fires with abundant spotting, where new ignitions ahead of the main fire multiply the burned area faster than the front alone could achieve.

The Haypress Fire—observed with abundant spotting—exhibits quadratic growth, its area expanding like $t^2$ as spot fires ignite ahead, then ignite ahead of themselves. The Goodview Fire, observed with minimal spotting, grows linearly. The difference in growth rate isn't trivial. A fire growing quadratically will, over time, consume vastly more area than one growing linearly—assuming similar spread rates at the front. Understanding which growth mode a fire is in, and what determines that mode, is central to predicting fire behavior.

The Physics of a Flying Ember

An ember begins its life when heat, combustion, and mechanical stresses break off a piece of vegetation—bark, leaves, twigs—or building material from a structure. Once detached, it faces a journey determined by forces as old as fluid dynamics: drag from the moving air, gravity pulling it downward, and its own terminal velocity, the speed at which gravitational acceleration is exactly balanced by aerodynamic drag.

But embers aren't passive particles. They burn in flight, their composition changing as pyrolysis converts solid material to flammable gases. Their shape changes, their mass changes, their aerodynamic properties change. An ember that starts as a chunk of bark may burn down to a smaller, lighter fragment that travels further, or it may break apart entirely. The combustion characteristics of embers in flight—their shape, the mechanical forces they experience, their changing composition as they pyrolyze and lose mass—are crucial to understanding their ultimate role in igniting new fuels or structures.

The atmosphere plays a dual role. At the surface, in the lowest meters of the atmospheric boundary layer, embers can be pushed by wind horizontally along the ground—a mode the authors call "ember wash." In this mode, embers don't rise significantly but instead roll, skip, and glide along the surface, sometimes gathering in eddies behind obstacles like trees or buildings. This surface mode may be the dominant transport mechanism when no lofting occurs initially, whether because embers are too heavy, the fire plume is not strong enough, or turbulence is intermittent. Embers in this mode can still travel considerable distances over flat terrain, and they pose particular dangers near structures, where they can accumulate against walls, in gutters, or under eaves, finding the kind of protected, often combustible spaces where ignition becomes likely.

Above the surface, embers can be lofted by the fire's own convection—the buoyant plume of hot air rising from the flames. This upward motion injects embers into the stronger winds of the free atmosphere, where they can travel tens of miles before falling back to earth. The lofted mode is what makes spotting so dangerous: embers released from the main fire can leapfrog entire fire perimeters, igniting spot fires in unburned areas far ahead of any containment efforts.

The turbulence that characterizes the atmospheric boundary layer adds further complexity. Wind doesn't flow smoothly; it churns with eddies of varying sizes, from the largest atmospheric structures down to the tiny swirls that dissipate energy as heat. These turbulent eddies can eject embers from the surface to higher levels, creating what might effectively be a thicker "boundary layer mode" of near-surface transport. Various transport processes may all operate simultaneously in the complex three-dimensional turbulent flow near the surface, and modeling these interactions requires sophisticated techniques.

The authors use numerical simulations with the CM1 atmospheric model to visualize these processes. The left panel of one figure shows the background wind conditions—wind speed and direction at various heights—while the right panel shows the simulated smoke concentration and ember particle density distribution. Smoke, being lighter than air, rises and spreads downstream, but embers, being heavier, follow more complex trajectories that include settling, lofting, and interactions with turbulent structures.

The full physics of ember transport, including combustion, lofting, settling at terminal velocity, and turbulent wind advection, can be captured in computational fluid dynamics (CFD) models. These simulations track individual ember "particles" as they move through a modeled wind field, accounting for the changing forces on each ember as its properties evolve. While computationally expensive—requiring massive parallel computing to simulate even a single fire event—these models provide the most complete representation of ember behavior currently available.

Statistical Models for Where Embers Land

But physics-based simulations aren't the only way to understand ember transport. The authors also explore statistical approaches that treat the ember landing pattern as a probability distribution. These statistical models sacrifice detail about individual ember trajectories in exchange for insight about aggregate behavior—which matters for risk assessment and emergency planning.

Researchers have described ember landing patterns using exponential distributions, lognormal distributions, and other empirical forms. Some models suggest "short-range" lognormal and "long-range" Poisson distributions, meaning that at different distances from the fire, different mathematical forms better capture where embers tend to land. A Poisson distribution describes events that happen randomly and independently, such that the number of events in a fixed interval follows a predictable pattern—so if ember landing is essentially random beyond a certain distance, a Poisson model might describe how many embers land in a given area.

An influential analysis by Storey et al. examined numerous factors influencing spotting and concluded that fire area is the most important predictor of spotting distance, with weather, topography, and fuels making secondary contributions. This makes intuitive sense: a larger fire produces more embers, has a stronger plume, and covers more of the landscape, all of which increase the probability that embers will travel far and find receptive fuel.

The authors also connect fire growth patterns to ember transport. As noted earlier, fires exhibiting quadratic area growth tend to have abundant spotting—spot fires ahead of the main front that multiply the burned area faster than the front alone could achieve. This suggests a feedback loop: more spotting leads to more fire area, which leads to more embers, which leads to more spotting. Understanding this feedback is crucial for predicting when a fire might transition from a manageable burn to an exponentially growing catastrophe.

The Surface Mode: Ember Wash

While much attention focuses on embers lofted high into the atmosphere, the surface transport mode deserves its own consideration. In the "ember wash" regime, embers travel along or near the ground, pushed by wind but not rising significantly. This mode may dominate when embers are heavy, when the fire plume is weak, or when atmospheric conditions suppress lofting.

Quaife and Speer developed an idealized statistical approach to model ember transport processes that control fire area growth from spotting and surface modes of transport. This approach represents a middle ground between simple physical intuition and full CFD simulation: idealized enough to analyze mathematically, but capturing enough of the real physics to provide genuine insight.

The Camp Fire provides a vivid example of surface ember transport in action. In aerial photographs, embers are visible flowing around trees and along the ground, their motion shaped by the local wind field and the obstacles around them. In the WUI—wildland-urban interface—where structures replace trees as the dominant features, embers can accumulate against buildings, pool in corners, and find ignition pathways that might otherwise be blocked. In mass fires throughout history, from wartime bombings to the great urban conflagrations of the early 20th century, burning debris blown downwind and landing on combustible fuels has multiplied fire spread in what becomes effectively an "urban" regime of spotting.

Full-Physics Simulations: The Dixie Fire

To test whether the theoretical framework holds up against real-world complexity, the researchers applied it to the 2021 Dixie Fire—a massive wildfire that burned nearly 1 million acres in northern California over more than three months. They simulated the fire using a coupled fire-atmosphere model and compared the simulated ember transport to observed fire behavior.

The simulation tracked ember particles released from the simulated fire, following their three-dimensional trajectories as they moved through the modeled atmosphere. The colors in one figure indicate the release height of each ember—ranging from near the surface to high in the atmospheric boundary layer. The trajectories reveal how embers from different heights travel differently: those released near the surface tend to follow more horizontal paths, while those lofted high follow more complex trajectories shaped by the stronger winds aloft.

The right panel of the figure shows the spatial distribution of ember landing locations, with colors representing how long each ember spent in the atmosphere before landing—the "residence time." Embers that spend more time aloft tend to travel further, but they also have more time to burn out or break apart. Understanding the distribution of residence times is crucial for predicting spotting distances and the probability of ignition upon landing.

The Dixie Fire simulation demonstrates that the modeling framework can reproduce key features of real-world fire behavior—including the tendency for embers to travel far, to spot ahead of the main fire front, and to create new ignitions in patterns that are statistical rather than deterministic. The match isn't perfect; real fires involve uncertainties that no model fully captures. But the framework provides a way to think about the problem systematically, to identify which variables matter most, and to make predictions that can be tested against observations.

Why This Changes Things

The mathematics of fire spread might seem like an abstraction, but its implications are profoundly practical. Every year, wildfires kill people, destroy homes, and reshape landscapes. The fires that do the most damage are often those that outpace evacuation, that spot ahead of fire lines, that create multiple ignitions where responders can only be in one place at a time. Understanding the physics of ember transport doesn't just satisfy scientific curiosity—it could save lives.

Consider what the framework offers. At the most basic level, it provides a way to think about fire spread that's more accurate than simple intuition. Many people, when they imagine wildfire, picture flames advancing like a wave, burning everything in a continuous front. But real wildfires don't advance continuously; they advance in jumps, with spot fires igniting ahead and multiplying. The elliptical model of fire growth—embedded in operational systems like FARSITE—captures some of this complexity, but the authors show that it doesn't capture everything. Fires with abundant spotting grow differently than fires without it, and the statistical models for ember transport provide a way to account for this difference.

The finding that fire spread rates follow Pareto distributions with heavy tails is particularly important. Most of the time, fires spread at moderate rates—about 0.5 meters per second on average, or roughly walking pace. But the tail of the distribution contains events far beyond this typical range: fires spreading at 10 meters per second or faster. These extreme events are rare, but they're the ones that overwhelm containment, that trap evacuees, that create the conditions for catastrophe. Understanding that these extreme events are not just random noise but part of a predictable statistical pattern changes how we should prepare for them.

The fire growth scaling—whether a fire grows linearly or quadratically—has implications for fire management and for modeling. A fire that grows quadratically will, over time, become vastly larger than one that grows linearly, even if both start with similar area and spread rates. If we can predict which growth mode a fire is in, and what factors determine the mode, we can better allocate resources, set evacuation perimeters, and anticipate the ultimate size of a fire. The connection between quadratic growth and spotting suggests that fires with abundant ember production may be more likely to undergo exponential expansion—and that early detection and suppression of spot fires might be especially valuable.

For the wildland-urban interface, where homes intermingle with wildland fuels, the surface transport mode deserves particular attention. Embers traveling along the ground can accumulate against structures, find ignition pathways that wouldn't exist in pure wildland settings, and create the "urban" regime of fire spread where burning debris multiplies destruction. The framework provides a way to think about WUI fire risk that's more nuanced than the simple "fire meets house" model that often underlies public perception and policy.

What's Next

The authors acknowledge that their framework, while comprehensive, has limitations. True model validation remains challenging because fire observations are inherently difficult: fires are chaotic, conditions change rapidly, and deploying instruments in the path of a wildfire is dangerous. Satellite data provides valuable coverage but has its own uncertainties, and false attributions of fire detections can contaminate datasets. "Great care must be taken interpreting observations and experimental results and comparisons to models," they write—a reminder that science doesn't progress by ignoring uncertainty but by characterizing it.

Several open questions remain. How do the statistical patterns of ember landing change under different weather conditions? How does topography—steep canyons, ridges, valleys—affect ember transport? What determines the probability that an ember landing on a given fuel type will actually ignite it? The ember source term—how many embers are produced per unit area of fire, per unit fuel type, per unit weather condition—remains incompletely characterized. Fire area appears to be the most important predictor of spotting, but the secondary contributions from weather, topography, and fuels are not fully quantified.

The authors also point toward connections with wave phenomena in the atmosphere. Just as ocean waves can drive a Stokes drift that moves floating objects, atmospheric waves from various sources might influence ember transport. Fire plume turbulence can amplify ember transport in ways that simple models miss. These are frontiers where the physics is still being worked out.

Perhaps most importantly, the framework invites integration with operational fire management systems. The semi-empirical fire spread models—Rothermel, McArthur, and their descendants—have provided the engine for systems like FARSITE, CAWFE, WRF-Fire, and others that are used daily by fire managers. If the statistical models for ember transport can be simplified enough to run in real time, they could enhance these systems' ability to predict spotting and to anticipate where new ignitions might occur. This is a significant challenge: operational systems must run fast enough to be useful, and adding complexity can slow them down. But the hierarchy of models presented here—from simple idealized cases to full CFD simulations—suggests a path forward, with different levels of complexity appropriate for different purposes.

The science of fire spread has come a long way from the days when fire behavior was predicted primarily from tables and intuition. The framework in this paper represents a synthesis of decades of research into something coherent and actionable. It won't answer every question, and it won't make fire prediction perfect. But by understanding fire not as a monolithic wall of flame but as a cascade of embers, each one following physical laws, the authors have given us a new way to think about an old problem. And that new way of thinking might, in time, help us live with fire a little more safely.