Drag-Based Model (DBM) Overview

• Drag-Based Model (DBM) is a class of models that uses a dominant drag law to describe CME propagation and unresolved obstacles in fluid dynamics.
• It employs analytical solutions and empirical parameter estimation to predict arrival times, speeds, and hit probabilities for space weather events.
• Extensions of DBM integrate ensemble forecasting, geometric variations, and data assimilation to enhance forecast accuracy in both heliospheric and environmental applications.

Drag-Based Model (DBM) denotes a class of drag-centered models in which the dominant interaction is represented through a drag law rather than through a full resolution of all forces or geometry. In space weather, the term most commonly refers to an analytical, physics-based model for the heliospheric propagation of coronal mass ejections (CMEs), based on the observational fact that slow CMEs accelerate whereas fast CMEs decelerate as they interact with the ambient solar wind, and typically applied beyond about $15$–$20\,R_\odot$ from the Sun (Dumbovic et al., 2021, Žic et al., 2015). In environmental and urban CFD, related drag-based formulations are used to represent unresolved trees, waves, and other obstacles through momentum-sink or surface-drag terms rather than explicit geometry (Tokiwa et al., 24 May 2026, Aiyer et al., 2021).

1. Standard heliospheric DBM

The standard CME DBM assumes that, once the eruption has propagated sufficiently far from the Sun, the dominant force is the MHD analog of aerodynamic drag. Its core acceleration law is

$a(t)=-\gamma\,(v(t)-w)\lvert v(t)-w\rvert,$

where $v$ is the CME speed, $w$ is the ambient solar wind speed, and $\gamma$ is the drag parameter. The model expresses the tendency of the CME speed to asymptotically approach the solar wind speed: fast CMEs decelerate, slow CMEs accelerate, and the drag acts to reduce the speed difference between CME and background flow (Dumbovic et al., 2021).

For constant $w$ and $\gamma$, the model admits analytical solutions. One common form is

$$R(t)=\frac{S}{\gamma}\ln[1+S\gamma(v_0-w)t]+wt+R_0,\qquad v(t)=\frac{v_0-w}{1+S\gamma(v_0-w)t}+w,$$

with $S=\mathrm{sign}(v_0-w)$, initial speed $20\,R_\odot$0, and initial distance $20\,R_\odot$1 (Dumbovic et al., 2021). Equivalent sign-explicit forms are also used in later parameter-inference work (Napoletano et al., 2022). Predictions for time of arrival and speed of arrival at a target, usually $20\,R_\odot$2, are obtained by solving these expressions at the target distance (Napoletano et al., 2022).

The drag parameter is written as

$20\,R_\odot$3

where $20\,R_\odot$4 is a dimensionless drag coefficient, $20\,R_\odot$5 is CME area, $20\,R_\odot$6 is solar wind density, $20\,R_\odot$7 is CME mass, and $20\,R_\odot$8 is virtual mass (Abu-Shaar et al., 29 Sep 2025). In practice, because of observational uncertainty, $20\,R_\odot$9 is often set empirically; one overview reports a typical range of $a(t)=-\gamma\,(v(t)-w)\lvert v(t)-w\rvert,$0–$a(t)=-\gamma\,(v(t)-w)\lvert v(t)-w\rvert,$1 and a default value of $a(t)=-\gamma\,(v(t)-w)\lvert v(t)-w\rvert,$2 in available DBM tools (Dumbovic et al., 2021).

A recurrent qualification in the CME literature is that the model is physically applicable to the CME magnetic structure, although it is often used as a proxy for shock arrival (Dumbovic et al., 2021). A common misconception is therefore to treat the standard DBM as a complete description of CME dynamics. The cited work instead presents it as a drag-dominated asymptotic model whose validity depends on distance from the Sun and on the neglect of Lorentz forces, gravity, and other non-drag effects.

2. Geometrical realizations and operational tool families

The analytical core of the DBM has been embedded in several geometrical and ensemble variants. A widely used overview identifies five versions developed at Hvar Observatory: a basic 1D DBM; an advanced 2D self-similar cone DBM; a 2D flattening cone DBM; DBEM, an ensemble version of the 2D flattening cone DBM; and DBEMv3, an ensemble version that creates CME ensembles based on the input uncertainties (Dumbovic et al., 2021).

Version	Geometry or ensemble assumption	Main output
1D DBM	Point or concentric arc	Arrival time and speed
2D self-similar cone DBM	Cone with self-similar expansion	Arrival and hit status
2D flattening cone DBM	Cone with independently propagated front elements	Arrival and hit status
DBEM	Ensemble 2D flattening cone DBM	Arrival probability, times, speeds, confidence intervals
DBEMv3	Ensemble 2D flattening cone DBM with uncertainty-generated inputs	Arrival probability, times, speeds, confidence intervals

The geometric extensions matter because off-apex impacts, CME width, and front flattening strongly affect hit–miss classification and arrival conditions. The 2D flattening cone DBM propagates each plasma element of the front independently, allowing the front to flatten over time as different segments experience different drag histories (Dumbovic et al., 2021). This is a more realistic treatment for fast or asymmetric CMEs than a purely self-similar cone.

A further geometric development is ElEvoHI, which combines an elliptic conversion method (ElCon), DBM fitting, and the Ellipse Evolution Model (ElEvo). Using STEREO heliospheric imager observations, it assumes an elliptical CME front with adjustable angular width and adjustable radius of curvature, fits the DBM to quantify deceleration or acceleration during propagation, and then propagates the fitted ellipse to a target (Rollett et al., 2016). For a sample of 21 CMEs, ElEvoHI improved the arrival time forecast by about 2 hours to $a(t)=-\gamma\,(v(t)-w)\lvert v(t)-w\rvert,$3 hours and the arrival speed forecast by $a(t)=-\gamma\,(v(t)-w)\lvert v(t)-w\rvert,$4 to $a(t)=-\gamma\,(v(t)-w)\lvert v(t)-w\rvert,$5, depending on the ellipse aspect ratio assumed (Rollett et al., 2016).

3. Event-specific fitting, statistical parameterization, and data assimilation

One of the central methodological issues in DBM practice is the specification of $a(t)=-\gamma\,(v(t)-w)\lvert v(t)-w\rvert,$6 and $a(t)=-\gamma\,(v(t)-w)\lvert v(t)-w\rvert,$7. In the standard formulation, these are often chosen from statistical studies or ad hoc assumptions. An important extension is DBM fitting by least squares, in which CME observations over a given distance range are used to evaluate the most suitable model input parameters for a given CME (Žic et al., 2015). The fitting objective is written as

$a(t)=-\gamma\,(v(t)-w)\lvert v(t)-w\rvert,$8

and the procedure reports fit-quality statistics including standard deviation, coefficient of variation, and coefficient of determination $a(t)=-\gamma\,(v(t)-w)\lvert v(t)-w\rvert,$9 (Žic et al., 2015). The same work emphasizes that new observations can be assimilated by repeating the fit as the dataset expands, and that segmented fitting can accommodate changing solar wind conditions or CME–CME interactions.

A complementary line of work estimates statistical distributions for DBM input parameters from past CME–ICME events. Using inversion of the DBM equations over more than 500,000 inversion trials, one study found that the solar wind speed $v$0 is bimodal and well fit by the sum of two Gaussians, whereas $v$1 is well fit by a lognormal distribution with distinct values and shapes depending on whether CMEs are accelerated or decelerated by the solar wind (Napoletano et al., 2022). This result supports ensemble sampling schemes already used in probabilistic DBM implementations, while also indicating that separate $v$2 distributions for accelerating and decelerating events may improve physical realism (Napoletano et al., 2022).

More recently, HELIOPANDA integrates the DBM with iterative parameter estimation and Kalman filter assimilation. It estimates $v$3 and $v$4 directly from position and speed measurements by linearizing the analytical DBM solutions with respect to those parameters, iterating up to 25 times, and using a grid of initial guesses with physically reasonable bounds $v$5 and $v$6 (Abu-Shaar et al., 29 Sep 2025). On 4,480 synthetic CME profiles, the framework reconstructed DBM input parameters accurately; with a single virtual spacecraft 30 million km from the Sun, it achieved arrival-time errors as low as 0.6 hours for a 600 km/s CME and 1 hour for a 2500 km/s CME, and with 160 simulated hourly measurements it yielded Earth and Mars arrival-time predictions within 1–2 hours (Abu-Shaar et al., 29 Sep 2025). This places DBM within a broader sequential-estimation and real-time forecasting architecture rather than treating it solely as a closed-form propagator.

4. Ensemble forecasting, forecast skill, and recognized limitations

The ensemble extension of the DBM, DBEM, treats CME input parameters and model parameters as uncertain quantities and propagates an ensemble of realizations. Because the DBM is analytical, ensemble computations remain fast: one evaluation reports a DBM runtime of $v$7 s, and DBEM can perform multiple runs in almost real time, within a few minutes (Čalogović et al., 2021, Dumbović et al., 2018). The ensemble output provides a probability of arrival, distributions of arrival times and speeds, medians as most likely values, and $v$8 confidence intervals (Dumbović et al., 2018, Čalogović et al., 2021).

Evaluation against observed events shows that DBEM is competitive with more complex heliospheric models, but also reveals persistent biases. In one comparison against ensemble WSA-ENLIL+Cone, DBEM achieved $v$9 hours, $w$0 hours, and $w$1 hours, compared with ENLIL errors of $w$2 hours, $w$3 hours, and $w$4 hours (Dumbović et al., 2018). A later evaluation of DBEMv3 on 146 CME–ICME pairs reported $w$5 h and $w$6 h, together with a clear bias toward negative prediction errors, where fast CMEs are predicted to arrive too early (Čalogović et al., 2021).

The cited literature is consistent on the principal limitations. The model assumes constant $w$7 and $w$8 beyond a certain heliocentric distance, simple CME geometry, and the absence of time-dependent or external forces other than drag (Dumbovic et al., 2021). It does not intrinsically describe nonhomogeneous solar wind, rapid early deceleration, CME–CME interaction, or the distinction between the CME magnetic structure and the leading shock (Dumbović et al., 2018, Čalogović et al., 2021). Fast CMEs are a particularly difficult regime: both DBEM and ENLIL tend to predict earlier arrivals than observed, and the DBM literature attributes this partly to model physical limitations and partly to possible overestimation of CME launch speed from observations (Dumbović et al., 2018).

5. Extended DBM and physics-informed machine learning

The most direct modification of the standard heliospheric DBM is the Extended Drag-Based Model (EDBM), which adds a constant extra acceleration term $w$9 to represent forces other than aerodynamic drag:

$\gamma$0

For $\gamma$1, the model admits residual acceleration; for $\gamma$2, it allows additional deceleration; and for $\gamma$3, it reduces to the standard DBM (Rossi et al., 2024). The extension is motivated by observations in which CME speed evolution is incompatible with drag-only dynamics. The paper emphasizes that magnetic forces, pressure gradients, internal CME restructuring, residual solar gravity, and magnetic or plasma pressure imbalances can all contribute to a nonzero $\gamma$4, and it shows that the asymptotic speed shifts from $\gamma$5 in the standard DBM to $\gamma$6 in the extended case (Rossi et al., 2024).

EDBM also changes the scope of model-guided machine learning. A recent framework generalizes a physics-driven AI approach from the classical DBM to the EDBM, explicitly to include CME events whose interplanetary dynamics are incompatible with the assumptions of the standard DBM (Lampani et al., 22 Dec 2025). The training loss combines an empirical travel-time term with a physics-based term enforcing agreement between the predicted arrival time and the EDBM radial position at $\gamma$7. The study reports travel-time prediction accuracy comparable to state-of-the-art methods, approximately 13-hour MAE and typically within 16% relative error, and notes that EDBM-based coverage extends to events representing up to 25% of the observed test set that would otherwise fall outside the classical DBM framework (Lampani et al., 22 Dec 2025). It also introduces a multiclass logistic-regression classifier for EDBM propagation regimes, with accuracy up to 79%, as part of a possible near-real-time operational pipeline (Lampani et al., 22 Dec 2025).

A plausible implication is that DBM is evolving from a single analytical formula into a modular forecasting framework. In this framework, closed-form drag physics remains the backbone, but fitting, inference, ensemble sampling, regime classification, and assimilation increasingly determine practical forecast quality.

6. Drag-based formulations in CFD, micrometeorology, and air–sea interaction

Outside heliospheric propagation, drag-based modeling appears as a parameterization strategy for unresolved geometry. In urban micrometeorology, trees are commonly represented as porous media because explicit tree geometry is computationally prohibitive. Conventional porous-media tree models prescribe a constant drag coefficient even with heterogeneous area-density distributions, which limits transferability across inflow conditions and increases grid-resolution sensitivity (Tokiwa et al., 24 May 2026). A recent fractal-based variable-drag framework replaces constant drag with cell-wise coefficients

$\gamma$8

where $\gamma$9 is a cell-effective branching order inferred from the in-cell self-similarity measure $w$0, and $w$1 is a cell-effective Reynolds number (Tokiwa et al., 24 May 2026). The resulting porous-media sink term is

$w$2

Across six grid resolutions $w$3 and three inflow Reynolds numbers, the model reduced the standard deviation of the drag effect across grid resolutions by up to 60% relative to a general constant-$w$4 model and by about 30% relative to an advanced constant-$w$5 model with spatially heterogeneous PAD; it also reproduced the inflow-velocity dependence of bulk drag without empirical retuning and produced plausible aerodynamic responses including velocity deficit, bypass flow, and wake recovery (Tokiwa et al., 24 May 2026).

A related use of drag-based modeling appears in large-eddy simulation of wind–wave interaction. There, a sea surface-based hydrodynamic drag model specifies the pressure-based surface drag felt by the wind due to the waves using local wave geometry, relative wind–wave speed, and a drag coefficient that depends on wave steepness (Aiyer et al., 2021). For unresolved vertical waves, the drag force per unit volume is formulated as

$w$6

with

$w$7

The model was validated against laboratory experiments across a wide range of wave ages and wave steepnesses, showed good agreement in mean velocity profiles and form stress, captured spatial and temporal variability because it is phase-aware, and achieved these results at an estimated $w$8 times smaller computational cost than phase-resolved LES (Aiyer et al., 2021).

These non-heliospheric examples do not define the canonical CME DBM, but they clarify the broader methodological meaning of drag-based modeling. In each case, unresolved structure is replaced by a drag law tied to local morphology, kinematics, or both. This suggests a common pattern across applications: the quality of a drag-based model depends less on the existence of a drag term itself than on how event-specific, geometry-aware, or flow-aware the drag closure is.