A Discrete Particle Modeling Framework for Exploring the Evolution of Aeolian Sediment Transport on Moist Sand Surfaces

Abstract

Aeolian sand transport on beaches is strongly affected by surface moisture, but its influence on transport evolution remains poorly understood. We present a novel discrete particle modeling framework to systematically investigate how moisture from liquid bridges affects the development of transport toward steady state after initiation. Moist sediment particles are modeled using a particle-based approach with evolving liquid bridges coupled to a one-dimensional airflow solver. The model captures realistic grain-scale collision dynamics under moist conditions and reproduces key features of aeolian transport in the dry limit. Simulations reveal two distinct behaviors: In steady state, the transport rate remains insensitive to moisture as lower saltation concentrations are compensated by higher saltation velocities; in the transient phase, however, increasing moisture prolongs the growth phase and delays the peak in transport rate. This delay arises because sand concentration peaks increasingly later than mean saltation velocity as moisture content increases. By projecting the temporal model results into a spatial representation, the position of peak transport is found to scale linearly with wind shear velocity, showing greater sensitivity at higher moisture levels. A preliminary test suggests that evaporation is essential for the initial grain dislodgement by wind alone but is unlikely to affect steady-state transport. This study explains the longer fetch distances observed on moist beaches before transport equilibrates and provides a physics-based tool for predicting sediment transport under varying moisture conditions.

1. Introduction

Surface moisture is known to complicate the process of aeolian sediment transport on sandy beaches by increasing the threshold wind velocity [1,2,3,4,5], altering the transport rate [3,5,6,7,8], and varying the vertical transport profile [7,8,9]. Despite the advances in understanding the role of moisture on transport locally, accurately predicting long-term transport from the beach to foredunes remains a significant challenge. The spatially inhomogeneous surface conditions, induced by the cross-shore moisture gradient, interfere with the way aeolian transport evolves downwind [9,10,11]. Extensive research has examined the evolution of sediment transport for dry conditions in both wind-tunnel and field settings [12,13,14,15,16]. However, the evolution of transport on moist surfaces remains poorly understood.

The limited field evidence suggests that the critical fetch distance required to reach the peak transport rate or equilibrium is significantly longer on moist surfaces [6,11,15,16]. Specifically, in [16], an overshoot pattern of the transport rate prior to steady-state transport was observed on both dry and moist surfaces, while the growth phase before overshoot was significantly extended on moist surfaces. Nevertheless, the physical mechanism behind these phenomena remains unclear. Furthermore, field studies are complicated by highly variable surface conditions, driven by natural processes such as precipitation and evaporation, making systematic investigations into the effects of moisture impractical. Meanwhile, wind-tunnel experiments have primarily focused on the unsaturated phase of transport, often with tunnel lengths too short to allow the system to reach transport saturation [1,2,5,7,17]. To address these limitations, we will employ a novel grain-scale model to provide new insights into these experimentally challenging phenomena.

Discrete particle modeling (also known as DPM), although computationally expensive and limited to small systems, offers precise control over parameters like particle positions and liquid properties, enabling systematic exploration of variables that are difficult to manage in experiments or field studies [18,19]. It also provides detailed insights into particle interactions and liquid distributions, offering a depth of analysis beyond conventional methods. Previously, DPM simulations have been successfully used to investigate sediment transport across various environments [20,21,22]. In particular, simulations with periodic boundary conditions enable the reintroduction of particles into the system after they leave the domain, facilitating continuous transport until a steady state is reached [23]. More recently, the method has been extended to incorporate general inter-particle cohesion by introducing additional forces at particle contacts [24,25,26,27].

In the simulations, the cohesive strength of liquid bridges is quantified by the cohesion number, defined as the ratio of the maximum capillary force to the gravitational force. This parameter is influenced by surface tension, which varies for different liquids, and plays a critical role in affecting the transport process [24,27]. However, the dynamics of liquid bridge volumes, governed by the water content, also strongly influence how the cohesive force evolves with the separation distance between grains [28]. This aspect could significantly impact transport behavior but has received limited attention in previous studies. Earlier research often assumed that cohesive forces are uniformly applied across all particle contacts [24,26,27]. In contrast, we demonstrate that the distribution of liquid bridges is typically non-uniform due to liquid migration across the particle contact network. To address this, our work focuses on the effects of water-based moisture under a fixed cohesion number while varying liquid bridge volumes, accounting for the redistribution of liquid between inter-particle bridges and that adhering to particle surfaces. This advancement provides a robust framework for exploring the complex interplay between moisture and sediment transport at the micro-mechanical level.

The objective of this study is to systematically investigate the impact of moisture from liquid bridges with varying volume content on the evolution of aeolian sediment transport over a time scale of seconds using a novel DPM-based model. The moisture content in the bed will be varied through the volume of inter-particle liquid bridges systematically per model run, isolating the effect of different levels of moisture content in the bed from factors that might change moisture content over time, such as evaporation or groundwater dynamics. Our approach is the first to couple sediment particles, evolving liquid bridges, and airflow dynamics. The paper is organized as follows. First, we introduce our model, which incorporates a liquid bridge model and a one-dimensional airflow model that is coupled to the particle dynamics. We then validate the model in two steps to ensure its accuracy and reliability. Next, we apply the model to systematically study the influence of moisture on steady-state transport and the evolution of transport.

2. Methodology

The transport model is implemented in the open-source software package MercuryDPM [29,30]. It consists of three elements: (1) grain motion, (2) liquid bridge and its evolution, and (3) airflow, respectively, each described in the subsequent sections.

2.1. Grain Motion Model

For this study, sediment grains are modeled as spherical particles for simplicity. Their movement is modeled by solving Newton’s law of motion:

$$m_i{\displaystyle \frac{\mathrm{d}{\overrightarrow{u}}_i}{\mathrm{d}t}}=m_i\overrightarrow{g}+{\overrightarrow{f}}_{\mathrm{air},i}+\sum _j({\overrightarrow{f}}_{\mathrm{c},i,j}+{\overrightarrow{f}}_{\mathrm{l},i,j})$$

(1)

$$I_i{\displaystyle \frac{\mathrm{d}{\overrightarrow{\omega }}_i}{\mathrm{d}t}}={\overrightarrow{r}}_{i,j}\times \sum _j({\overrightarrow{f}}_{\mathrm{c},i,j}+{\overrightarrow{f}}_{\mathrm{l},i,j})$$

(2)

$${\overrightarrow{r}}_{i,j}={\overrightarrow{c}}_{i,j}-{\overrightarrow{r}}_i$$

(3)

where ${\overrightarrow{u}}_i$ and ${\overrightarrow{\omega }}_i$ are the velocity and angular velocity of a certain particle with an index of i. $\overrightarrow{g}$ is the gravity acceleration, ${\overrightarrow{f}}_{\mathrm{air},i}$ is the air interaction force, and ${\overrightarrow{f}}_{\mathrm{c},i,j}$ and ${\overrightarrow{f}}_{\mathrm{l},i,j}$ are the contact force and the liquid-induced force due to the contact with another particle j, respectively. $m_i$ and $I_i$ are the mass and moment of inertia, ${\overrightarrow{r}}_{i,j}$ is the branch vector pointing from particle i to the contact point with particle j, and ${\overrightarrow{c}}_{i,j}$ and ${\overrightarrow{r}}_i$ are the position vectors of the contact point and particle i, respectively.

The contact force ${\overrightarrow{f}}_{\mathrm{c},i,j}$ is a combination of linear visco-elastic normal force and sliding friction in tangential direction [31]. During a collision between two neighboring particles, the contact force consists of a normal and a tangential component. The normal part is described by a linear spring force depending on the overlap and a damping force depending on the relative motion of the particles. The tangential part is similar to the normal part, differs in direction, but is controlled by the Coulomb friction limit.

The air phase is considered to exert the drag force and buoyancy force on the particles in this study, which follow the equations below.

$${\overrightarrow{f}}_{\mathrm{air},i}={\overrightarrow{f}}_{\mathrm{drag},i}+{\overrightarrow{f}}_{\mathrm{bouy},i}$$

(4)

$${\overrightarrow{f}}_{\mathrm{drag},i}={\displaystyle \frac{\pi }{8}}{\rho }_{\mathrm{a}}{d_i}^2{C}_{\mathrm{d}}|\overrightarrow{u}\left({\overrightarrow{r}}_i\right)-{\overrightarrow{u}}_i|(\overrightarrow{u}\left({\overrightarrow{r}}_i\right)-{\overrightarrow{u}}_i)$$

(5)

$${\overrightarrow{f}}_{\mathrm{bouy},i}=-{\displaystyle \frac{\pi }{6}}{d_i}^3{\rho }_{\mathrm{a}}\overrightarrow{g}$$

(6)

where ${\overrightarrow{f}}_{\mathrm{drag},i}$ and ${\overrightarrow{f}}_{\mathrm{bouy},i}$ are the drag force and the buoyancy force exerted on particle i by the surrounding air. ${\rho }_{\mathrm{a}}$ is the air density and $\overrightarrow{u}$ is the airflow velocity. $C_{\mathrm{d}}$ is the drag coefficient as a function of the particle Reynolds number ${\mathrm{Re}}_i=|\overrightarrow{u}\left({\overrightarrow{r}}_i\right)-{\overrightarrow{u}}_i|d_i/{\nu }_{\mathrm{a}}$, which depends on the velocity difference between the air phase and particle. Here, $d_i$ is the particle diameter and ${\nu }_{\mathrm{a}}$ is the air kinematic viscosity.

$$C_{\mathrm{d}}={\left(\sqrt{C_{\mathrm{d},\infty }}+\sqrt{{\mathrm{Re}}_{\mathrm{p},\mathrm{c}}/{\mathrm{Re}}_i}\right)}^2$$

(7)

where $C_{\mathrm{d},\infty }\simeq 0.5$ is the drag coefficient in the turbulent limit and ${\mathrm{Re}}_{\mathrm{p},\mathrm{c}}\simeq 24$ is the transitional particle Reynolds number between laminar and turbulent flow regimes [21].

2.2. Liquid Bridge Model and Its Evolution Rule

Particle collision is actively involved in the transport process not only at the sediment bed surface but also in mid air. The frequent collisions and contacts of moist sand particles cause the formation and rupture of liquid bridges, facilitating the redistribution of the interstitial liquid [32,33]. To account for this, we implemented a liquid migration rule that extends beyond the standard liquid bridge model applied to all the particle contacts, different from those in the previous studies [24,26,27]. This migration rule of liquid is given in [34,35], and implemented by [28] in MercuryDPM. The migration rule assumes that particles and liquid are two different entities in the system. The liquid is either attached to particles or to contacts as a liquid bridge, depending on the formation or breakage of contacts. Once a particle pair is subject to contact, the liquid migrates from the particle and forms a bridge with the neighboring particle. When the particles move away from each other, the distance S between them increases. The rupture distance $S_{\mathrm{r}}$ is an upper limit for the liquid bridge to exist. Once S reaches $S_{\mathrm{r}}$, the liquid bridge breaks and the liquid is redistributed to other bridges associated with the particles, or equally back to the particle surface if there is no other bridge [28].

The liquid bridge induces a capillary force proposed by [36] and a viscous lubrication force proposed by [37,38].

$${\overrightarrow{f}}_{\mathrm{l},i,j}={\overrightarrow{f}}_{\mathrm{l},\mathrm{cap},i,j}+{\overrightarrow{f}}_{\mathrm{l},\mathrm{vis},i,j}$$

(8)

where ${\overrightarrow{f}}_{\mathrm{l},\mathrm{cap},i,j}$ and ${\overrightarrow{f}}_{\mathrm{l},\mathrm{vis},i,j}$ are the capillary force and the lubrication viscous force, respectively, exerted on particle i by the liquid bridge formed with another particle j.

The capillary force within the rupture distance follows

$${\overrightarrow{f}}_{\mathrm{l},\mathrm{cap},i,j}={\displaystyle \frac{f_{\mathrm{l}},\mathrm{cap},\max }{1+S_{\mathrm{c}}(1.05+2.5S_{\mathrm{c}})}}{\overrightarrow{n}}_{i,j}\mathrm{for}S<S_{\mathrm{r}}\mathrm{with}{f}_{\mathrm{l}},\mathrm{cap},\max =2\pi \tilde{R}\gamma cos\theta$$

(9)

$$S_{\mathrm{r}}=(1+0.5\theta )·V_{\mathrm{lb},i,j}^{{\displaystyle \frac{1}{3}}}$$

(10)

in which $f_{\mathrm{l}},\mathrm{cap},\max$ is the maximum capillary force, ${\overrightarrow{n}}_{i,j}$ is the normal vector for a particle contact, $\gamma$ is the surface tension of the liquid, $\theta$ is the contact angle of the liquid with the particle surface, $V_{\mathrm{lb},i,j}$ is the volume of a liquid bridge between a particle pair, and $S_{\mathrm{c}}$ is the dimensionless separation distance calculated below.

$$S_{\mathrm{c}}=S\sqrt{{\displaystyle \frac{\tilde{R}}{V}}}$$

(11)

where $\tilde{R}$ is the effective radius of the particle pair with radii $R_i$ and $R_j$, respectively.

$$\tilde{R}={\displaystyle \frac{2R_i{R}_j}{R_i+R_j}}$$

(12)

The capillary force is conservative and only dissipates energy during the rupture event. Therefore, we also include the forces due to the viscous damping effects of the liquid. These viscous forces in both the normal and tangential directions are given as

$${\overrightarrow{f}}_{\mathrm{l},\mathrm{vis},i,j}^n={\displaystyle \frac{3\pi {\mu }_{\mathrm{l}}{\tilde{R}}^2}{2S}}{\overrightarrow{u}}_{\mathrm{r}}^n$$

(13)

$${\overrightarrow{f}}_{\mathrm{l},\mathrm{vis},i,j}^t=3\pi {\mu }_{\mathrm{l}}\tilde{R}\left({\displaystyle \frac{8}{15}}ln{\displaystyle \frac{\tilde{R}}{2S}}+0.9588\right){\overrightarrow{u}}_{\mathrm{r}}^t$$

(14)

in which ${\mu }_{\mathrm{l}}$ is the dynamic viscosity of the liquid and ${\overrightarrow{u}}_{\mathrm{r}}^n$ and ${\overrightarrow{u}}_{\mathrm{r}}^t$ are the relative velocities in normal and tangential contact directions. The viscous force is only valid above a minimum separation distance $S_{\mathrm{m}}$. Following [39], we take $S_{\mathrm{m}}={\displaystyle \frac{1}{100}}min(R_i,R_j)$.

2.3. Airflow Model

MercuryDPM is coupled with the FEM solver oomph-lib [30]. However, here, we directly implemented a one-dimensional airflow model, similar to the one in [40], in MercuryDPM for modeling the aerodynamics in the turbulent boundary layer. The governing equation for the airflow is a one-dimensional Reynolds-averaged Navier–Stokes equation, in which the flow velocity is assumed horizontal ($\overrightarrow{u}(t,\overrightarrow{r})=(u(t,z),0,0)$).

$${\rho }_{\mathrm{a}}(1-\varphi ){\displaystyle \frac{\partial u}{\partial t}}=(1-\varphi ){\displaystyle \frac{\partial {\tau }_{\mathrm{a}}}{\partial z}}-F_{\mathrm{d}}$$

(15)

where $\varphi$ and $F_{\mathrm{d}}$ are the particle volume fraction and the body force density that the particles exert on the air phase over a horizontal layer of the model domain, identified by a layer thickness of $dz$.

$$\varphi ={\displaystyle \frac{1}{L_{\mathrm{x}}{L}_{\mathrm{y}}dz}}\sum _{i\in z,z+dz}{\displaystyle \frac{\pi }{6}}{d_i}^3$$

(16)

$$F_{\mathrm{d}}={\displaystyle \frac{1}{L_{\mathrm{x}}{L}_{\mathrm{y}}dz}}\sum _{i\in z,z+dz}{\overrightarrow{f}}_{\mathrm{drag},i}$$

(17)

where $L_{\mathrm{x}}$ and $L_{\mathrm{y}}$ are the length and width of the model domain, respectively.

The airborne shear stress ${\tau }_{\mathrm{a}}$ can be related to the airflow velocity field through a Prandtl turbulent mixing length ℓ.

$${\tau }_{\mathrm{a}}={\rho }_{\mathrm{a}}({\nu }_{\mathrm{a}}+{\ell }^2|{\displaystyle \frac{\partial u}{\partial z}}|){\displaystyle \frac{\partial u}{\partial z}}$$

(18)

in which ${\nu }_{\mathrm{a}}$ is the air kinimetic viscosity. The turbulent mixing length ℓ is calculated through the differential equation proposed by [21].

$${\displaystyle \frac{\partial \ell }{\partial z}}=\kappa \left[1-exp\left(-\sqrt{{\displaystyle \frac{1}{R_{\mathrm{c}}}}\left({\displaystyle \frac{u\ell }{{\nu }_{\mathrm{a}}}}\right)}\right)\right]$$

(19)

where $\kappa \simeq 0.4$ is the von Karman’s constant and $R_{\mathrm{c}}=7$ is a dimensionless parameter [21,41].

The governing equation, Equation (15), is solved subject to the following boundary condition at the bottom of the domain, i.e., z = 0.

$$u\left(z\right)=0\mathrm{for}z=0$$

(20)

$$\ell \left(z\right)=\kappa z\left[1-exp\left(-{\displaystyle \frac{1}{R_{\mathrm{vD}}}}\left({\displaystyle \frac{z{u}_{\ast }}{{\nu }_{\mathrm{a}}}}\right)\right)\right]\mathrm{for}z\to 0$$

(21)

in which $R_{\mathrm{vD}}$ = 26 is the mixing length scaling constant suggested by [41].

At the top boundary of the model $z=L\mathrm{z}$, the air shear stress is prescribed by ${\tau }_{\mathrm{a}}={\rho }_{\mathrm{a}}{u_{\ast }}^2$, where $u_{\ast }$ is the shear velocity in the turbulent boundary in the absence of particles.

As the initial condition, the profiles of velocity and mixing length scale are prescribed such that it represents the steady-state solution when particles are fixed. In order to achieve this, we start from an initial guess for the velocity field in which the zero-velocity point is at the bed surface, defined as the highest vertical location with an average particle volume below 0.4. Then, we integrate Equation (15) while fixing the particles until the flow profile becomes constant.

Equation (15) is numerically solved through a semi-implicit finite difference scheme, similar to that in [40]. This is implemented using the linear algebra package Eigen in MercuryDPM.

2.4. Simulation Set-Up

The particle size is modeled with a mean diameter of $d_{\mathrm{mean}}=0.25$ mm, which is a common sand grain size found on sandy beaches [16,42]. To avoid crystallization of the particle bed, a normal distribution with a standard deviation of $d_{\mathrm{std}}=0.01$ mm is used, following other DPM studies on aeolian transport [43,44]. We simulate a domain of length $L_{\mathrm{x}}$ = 100$d_{\mathrm{mean}}$ and width $L_{\mathrm{y}}$ = 2$d_{\mathrm{mean}}$. To save computational efforts and to achieve the evolution of transport over time, periodic boundaries are applied to both x- and y-directions; i.e., the particles leaving the domain from one side will enter it with the same velocity from the other side, similar to the other DPM studies on aeolian transport [21,23,40]. A horizontal bottom wall is placed at z = 0, and an open boundary is applied to the top. Gravity is applied in z-direction. The particle bed is generated by pouring the particles from a certain height onto the bottom wall and letting them settle until stationary. To guarantee that the momentum of a saltating particle impacting the bed is completely dissipated within the bed, the bed should be at least 10d high [22]. Hence, 2715 particles are inserted, leading to a particle bed about 12$d_{\mathrm{mean}}$ high. The domain of the airflow solver has the same size as the bed in x- and y-directions but is 800$d_{\mathrm{mean}}$ high [40]. The flow velocity profile above the bed is shown in Figure A1 in Appendix A. A snapshot of the model simulation is shown in Figure 1c.

The model parameters for particle contact and air properties are selected from the literature and correspond to typical aeolian conditions [21,40], listed in

Table 1. Model parameters used in the DPM-based model.

Parameter	Symbol	Unit	Value
Particle contact parameters
Particle density	${\rho }_{\mathrm{p}}$	kg/m3	2650
Normal stiffness	$k_{\mathrm{n}}$	kg/s2	5000 $mg/d_{\mathrm{mean}}$
Normal damping coefficient	${\nu }_{\mathrm{n}}$	kg/s	4.7$m\sqrt{{\displaystyle \frac{g}{d_{\mathrm{mean}}}}}$
Tangential stiffness	$k_{\mathrm{t}}$	kg/s2	0.29$k_{\mathrm{n}}$
Tangential damping coefficient	${\nu }_{\mathrm{t}}$	kg/s	0.29${\nu }_{\mathrm{n}}$
Rolling stiffness	$k_{\mathrm{r}}$	kg/s2	0.4$k_{\mathrm{n}}$
Rolling damping coefficient	${\nu }_{\mathrm{r}}$	kg/s	0.4${\nu }_{\mathrm{n}}$
Sliding friction coefficient	${\mu }_{\mathrm{t}}$	-	0.5
Rolling friction coefficient	${\mu }_{\mathrm{r}}$	-	0.5
Liquid parameters
Surface tension	$\gamma$	kg/s2	0.0728
Contact angle	$\theta$	rad	$65·{\displaystyle \frac{\pi }{180}}$
Liquid dynamic viscosity	${\mu }_{\mathrm{l}}$	kg/m/s	0.001
Air parameters
Air density	${\rho }_{\mathrm{a}}$	kg/m3	1.225
Air kinematic viscosity	${\nu }_{\mathrm{a}}$	m2/s	1.46·10−5
Other parameters
Drag coefficient in the turbulent limit	$C_{\mathrm{d}},\infty$	-	0.5
Critical particle Reynolds number	$R{e}_{\mathrm{p}},\mathrm{c}$	-	24
Von Karman constant	$\kappa$	-	0.4
Mixing length scaling parameter	$R_{\mathrm{c}}$	-	7
Gravity acceleration	g	m/s2	9.81

. The liquid parameters are set to reflect the typical physical properties of water, in which the water–sand contact angle for the liquid bridge model is taken from [45]. The cohesion number in this work corresponds to $Co=f_{\mathrm{l},\mathrm{cap},\max }/mg$ = 113. The time step for solving the equations of motion of the particles is 1/50 of the collision time calculated by Equation (22),

$$t_{\mathrm{c}}=\pi /\sqrt{2k_{\mathrm{n}}/m-{({\nu }_{\mathrm{n}}/m)}^2}$$

(22)

in which m is mean particle mass, and $k_{\mathrm{n}}$ and ${\nu }_{\mathrm{n}}$ are normal stiffness and damping coefficient, respectively, defined according to [31]. The same time step is applied to the airflow solver since its critical time step is much larger than that for the particle dynamics.

The airflow conditions are controlled through the non-dimensionalized shear stress, i.e., the Shields number $\tilde{\mathrm{\Theta }}$. It is related to the shear velocity that determines the strength of the wind.

$$\tilde{\mathrm{\Theta }}={\displaystyle \frac{{\rho }_{\mathrm{a}}{u}_{\ast }^2}{({\rho }_{\mathrm{p}}-{\rho }_{\mathrm{a}})g{d}_{\mathrm{mean}}}}$$

(23)

The simulation parameters are listed in

Table 2. Parameters for the transport simulations.

Parameter	Symbol	Unit	Value
Mean particle diameter	$d_{\mathrm{mean}}$	mm	0.25
Standard deviation of particle diameter	$d_{\mathrm{std}}$	mm	0.01
Cohesion number	$Co$	-	113
Moisture content of the bed	$\mathrm{\Omega }$	%	0, 1, 5, 10, 20
Shields number	$\tilde{\mathrm{\Theta }}$	-	0.01, 0.02, 0.03, 0.04, 0.05, 0.06

. The Shields numbers selected for transporting the particles correspond to the shear velocities in the typical aeolian environment. The bed moisture content $\mathrm{\Omega }$ represents the water content by volume, defined as the ratio of the total liquid volume to the total solid volume and expressed as a percentage:

$$\mathrm{\Omega }={\displaystyle \frac{{\sum }_{i,j\in C}{V}_{\mathrm{lb},i,j}+{\sum }_i{V}_{\mathrm{lp},i}}{{\sum }_i{V}_{\mathrm{p},i}}}$$

(24)

in which $V_{\mathrm{lb},i,j}$ is the liquid bridge volume attributed to a particle pair in contact, C is the set of all particle pairs in contact, $V_{\mathrm{lp},i}$ is the liquid volume attributed to a particle, and $V_{\mathrm{p},i}$ is the volume of a particle. After the particle bed is generated, the liquid is added slowly to all the particle contacts in the form of liquid bridges until the prescribed volume is reached to minimize disturbances to the bed, as Figure 1a shows. The cohesive behavior of the simulated moist sediment bed has been validated through a direct shear test (see Appendix B for more details), and the test results are in line with [46]. The prescribed initial moisture profile reflects a sandy beach that is flooded by the tide and replenished from below by the groundwater. In field conditions, the gravimetric moisture content of the top sand layer is in the range of 0–10%, corresponding to 0–26.5% by volume for the density ratio in this study [47]. Here, the studied range of moisture content is limited to the pendular regime (0–20%), where the liquid bridge model is valid [48].

In this study, we aim to investigate the effect of moisture from liquid bridges on sediment transport dynamics beyond the initiation of transport. A preliminary test using our model suggests that initiating transport by wind on moist beds is significantly more difficult than on dry beds. It requires very high Shields numbers (10–20), far beyond the values corresponding to a typical aeolian environment, to overcome the maximum capillary force acting on the particles in the top layer (further discussed in Section 4). This indicates that relying solely on aerodynamic entrainment of the first particle to initiate transport is not realistic for sandy surfaces, where all grains are connected by liquid bridges. Other processes, such as evaporation, are thus required to free at least one particle from liquid bridges to realistically set the sand grains into motion by wind. Therefore, to circumvent the initiation of the first grains set into motion, we injected ten mobile particles into the saltation layer to initiate the transport through particle impact, as shown in Figure 1b, allowing us to focus on the subsequent evolution of transport. This situation, where upwind sediment flux already exists, can occur in field conditions, for example, at an interface between a dry sandy surface and a less erodible moist surface, as commonly observed on sandy beaches [49,50]. These conditions are particularly relevant for understanding the initiation and development of bedforms typically appearing on moist surfaces, such as sand strips. To make sure this initial particle flux does not affect the results qualitatively, we also ran simulations at $\tilde{\mathrm{\Theta }}$ = 0.06, with one and five particles injected, respectively. The detailed information about these injected particles is listed in

Table A1. Particle information for different initial conditions (the y position for all the initiating particles is 2.5 $\times {10}^{-4}$ m, and the particle velocity in y direction is 0).

Number of Initiating Particles (Nip)	Particle ID	Position $\times {10}^{-3}$ [m]		Velocity [m/s]		Diameter [mm]
		$\mathit{x}$	$\mathit{z}$	${\mathit{U}}_{\mathbf{p},\mathbf{x}}$	${\mathit{U}}_{\mathbf{p},\mathbf{z}}$	${\mathit{d}}_{\mathbf{p}}$
10	1	4	3.75	1	−1	0.28
	2	8	3.75	2	−2	0.25
	3	12	3.75	3	−3	0.22
	4	16	3.75	4	−4	0.19
	5	20	3.75	5	−5	0.16
	6	4	5	8	−8	0.16
	7	8	5	6.4	−6.4	0.19
	8	12	5	4.8	−4.8	0.22
	9	16	5	3.2	−3.2	0.25
	10	20	5	1.6	−1.6	0.28
5	1	4	3.9	2.10	−2.10	0.27
	2	8	4.1	3.34	−3.34	0.24
	3	12	4.4	3.90	−3.90	0.22
	4	16	4.6	3.44	−3.44	0.24
	5	20	4.8	2.13	−2.13	0.27
1	1	12	4.4	2.92	−2.92	0.25

in Appendix C.

Previous studies suggest that moisture evaporation can strongly influence aeolian sediment transport dynamics [51,52]. To assess the likely degree of evaporation during the time scales covered by our simulations, we estimated water loss under a range of environmental conditions, including relative humidity and air temperature (Appendix D). The calculations indicate that evaporation can be substantial on both the bed surface and saltating grains when conditions permit (see Section 4). Furthermore, in the intertidal zone, moisture dynamics are more complicated than evaporation alone. There will be regions of the beach where, even though evaporation takes place at its surface, its moisture is replenished via capillary rise from the groundwater that is close to the surface. Therefore, it will inevitably be very interesting to focus on how aeolian sediment transport dynamics beyond the onset of motion are affected by liquid bridges of varying volumes being present at the surface of the sediment bed, leaving these processes for future work.

2.5. Validation of the Modeling Framework

Particle impact is widely recognized as the dominant mechanism for entraining new grains into the saltation layer [53], in which the momentum restitution is influenced significantly by the presence of liquid bridges. Therefore, it is crucial to validate the liquid bridge model on the collision dynamics under moist conditions. In a previous study conducted by [54], an experiment was performed in which a glass bead (1.74 mm) collided with a flat surface both in dry and moist scenarios. In the moist scenario, a thin water layer is above the flat surface. The coefficients of restitution, defined as the rebound velocity over the impact velocity, were measured for various impact velocities. To perform the validation, we conducted a collision test involving a particle with the same diameter and a flat wall. The particle is with a liquid layer of the same thickness as that in the experiment in the moist scenario. The contact parameters are chosen such that the results for the dry particle rebound are not affected; the material properties used in this simulations are identical to those in the experiments [54,55]. Specifically, the particle density is 2500 kg/m³, the surface tension of water is 0.0728 kg/s², the dynamic viscosity is 0.001 kg/m/s, and the contact angle is assumed to be 35° [56]. The liquid on the particle is assumed to be distributed in a film of 100 μm. After conducting the collision tests, we compared the coefficients of restitution obtained from our simulations to the experimental results in both scenarios, which are depicted in Figure 2.

The results indicate that our model generally captures the coefficient of restitution quite accurately by including both capillary force and viscous lubrication force with the selected parameters. However, in the moist scenario, there is a slight underestimation observed in the low-velocity region and an overestimation in the high-velocity region. It is worth noting that [57] proposed a categorization scheme for particle–liquid collision dynamics, which states that energy dissipation is induced by different forces according to the value of capillary number ($Ca=U_{\mathrm{impact}}{\mu }_{\mathrm{l}}/\gamma$). In this tested range, the capillary force is dominant in the low-velocity region, but the high-velocity region corresponds to a regime where both capillary and viscous forces are influential. Consequently, the utilization of the same liquid force law in both regimes could contribute to the observed discrepancies. Nevertheless, the behavior of the collision dynamics, as indicated by our model, remains within an acceptable range. Thus, it is worth noting that including the viscous force is crucial for effectively capturing the dynamic behavior.

Validation of the Transport Model at the Dry Limit

To validate the transport model, simulations of transport on a dry sediment bed were performed. Figure 3a shows the evolution of transport rate over time with varying Shields numbers. Initially, the energetic impact particles and the excessive momentum in the air phase facilitate the entrainment of a substantial number of particles into the saltation layer, leading to a significant increase in the transport rate. However, due to the negative feedback resulting from the increased number of saltating particles, the momentum of the air phase gradually reduces, yielding a decrease in the transport rate. After this time period, including the growth and decline phase, the system eventually reaches a steady state when the momentum of saltating particles and the airflow field balance each other. The momentum of the airflow that changes over time is shown in Figure A1.

We extract the steady-state transport rates by averaging the values from the steady zone in the profile of the temporal transport evolution (see the definition of the steady zone in Section 3.1). Then, they are normalized using Equation (25) [21]. The plotted results in Figure 3b reveal that the simulated transport rate in the steady state, $Q_{\mathrm{steady}}$, exhibits a linear correlation with the Shields number, as claimed by [21]. They also align well with the measured data in wind tunnels and the empirical formulations [58,59,60,61].

$$\tilde{Q}={\displaystyle \frac{Q}{{\rho }_{\mathrm{p}}\sqrt{(\frac{{\rho }_{\mathrm{p}}}{{\rho }_{\mathrm{a}}}-1)g{d}_{\mathrm{mean}}^3}}}$$

(25)

Due to the periodic boundary conditions, the particle flow is spatially homogeneous. In order to compare the temporal evolution observed in our simulations to the spatial (downwind) evolutions observed in experiments, we related the time to the downwind travel distance of the saltation cloud [23]. By integrating the average saltation velocity $U_{\mathrm{sal}}\left(t\right)$ with time, through Equation (26), we calculated the downwind distance L. The average saltation velocity is taken from the coarse-grained particle velocity field in the saltation layer, defined as the vertical region where the particle velocity is higher than 0.1$u_{\ast }$ [20]. Figure 4a shows that the transport rate over downwind distance also displays a non-monotonous behavior, with an overshoot at a finite distance at first and a steady value sustaining at the end of the simulations. The evolution resembles the solution of a damped harmonic oscillator, as proposed by [14], with a strong correlation coefficient ($R^2$ = 0.97). It is worth noting that the prescribed initial airflow field in our model may lead to relatively high overshoots, particularly at high Shields numbers ($\tilde{\mathrm{\Theta }}>$ 0.04). This can result in less accurate fitting of the evolution curves under such conditions.

$$L\left(t\right)={\int }_0^t{U}_{\mathrm{sal}}d{t}^{\prime }$$

(26)

We defined the distance for the transport rate to reach its peak value as $L_{\mathrm{peak}}$ and normalized it by the drag length, as Equation (27) shows. Figure 4b shows that the simulated peak distance, $L_{\mathrm{peak}}$ increases with the Shields number, well in line with the experimental trend. According to [14], $L_{\mathrm{peak}}$ is not sensitive to the upwind particle flux for dry sediment transport. Thus, even though the initial particle flux in our simulations is different from in their experiments, the same dependency is satisfyingly reproduced.

$$\tilde{L}={\displaystyle \frac{L}{\frac{{\rho }_{\mathrm{p}}}{{\rho }_{\mathrm{a}}}{d}_{\mathrm{mean}}}}$$

(27)

3. Results

We performed 5 s transport simulations with various Shields numbers ($\tilde{\mathrm{\Theta }}$ = 0.01–0.06) and bed moisture contents ($\mathrm{\Omega }$ = 0–20%). We will first focus on the results of the steady part of the simulations in Section 3.1 and then present the results of the whole simulation time in Section 3.2.

3.1. The Steady State of Transport

We define the steady-state interval as the period from the first post-overshoot moment when the saltation concentration $C_{\mathrm{sal}}$ falls within ±15% of the mean computed over the final 1.5 s of the record to the end of the record, since $C_{\mathrm{sal}}$ saturates the latest among all the transport characteristics. The averaged values of several transport characteristics in the steady state of transport are calculated for all the Shields numbers and moisture contents. The saltation concentration ($C_{\mathrm{sal}}$) and mean saltation velocity ($U_{\mathrm{sal}}$) are taken as the height-integrated particle concentration in the saltation layer and the mean horizontal velocity of the saltating particles, respectively. The saltation height ($H_{50}$) is the height below which 50% of transport occurs. Depicted in Figure 5, the steady-state transport rate, $Q_{\mathrm{steady}}$, does not seem to be altered by moisture remarkably due to the opposing effects on the saltation concentration ($C_{\mathrm{sal},\mathrm{steady}}$) and velocity ($U_{\mathrm{sal},\mathrm{steady}}$), which counterbalance each other. $C_{\mathrm{sal},\mathrm{steady}}$ shows a decrease from dry to moist scenarios, suggesting that the ejection of new particles is restricted from a moist bed. In contrast, $U_{\mathrm{sal},\mathrm{steady}}$ and $H_{50,\mathrm{steady}}$ increase in the presence of moisture. The rise in $U_{\mathrm{sal},\mathrm{steady}}$ is consistent with the wind-tunnel observations by [62], which reported saltating particles’ velocities 1.5–2 times higher over moist sand than over dry sand. Likewise, the moisture-induced thickening of the saltation layer agrees with field measurements [8], which found nearly double $H_{50,\mathrm{steady}}$ on moist sand relative to dry under a similar range of wind shear velocity. Furthermore, among the moist bed conditions, the trend of these three characteristics with varying moisture content is not clearly apparent.

3.2. The Evolution of Transport

3.2.1. Temporal Evolution

The temporal evolution of the transport rate is shown in Figure 6a–f for different moisture and wind shear conditions. The same pattern is observed for all the Shields numbers and moisture contents: the transport rate initially exhibits an overshoot, then stabilizes at a significantly lower rate corresponding to the steady-state value, consistent with observations [13,14,16]. As the Shields number increases, the overshoot becomes increasingly apparent, and the peak value rises. As the bed moisture content increases, the transport rate reaches the peak much later, forming a more complex shape with a slower growth phase of transport rate before the actual overshoot, qualitatively matching the finding in field conditions [16].

To unravel the mechanism of this difference between dry and moist conditions, we show in Figure 6g,h the saltation concentration ($C_{\mathrm{sal}}$) and the mean saltation velocity ($U_{\mathrm{sal}}$) versus time at the highest Shields number $\tilde{\mathrm{\Theta }}$ = 0.06, where the difference is most visible. The results indicate that the peaks of $C_{\mathrm{sal}}$ and $U_{\mathrm{sal}}$ appear almost simultaneously for dry sediment. Conversely, under moist conditions, $U_{\mathrm{sal}}$ peaks first, followed by $C_{\mathrm{sal}}$. Specifically, for the most moist case ($\mathrm{\Omega }$ = 20%) marked by the red curve in Figure 6f–h, there is a clear lag between the peaks of $U_{\mathrm{sal}}$ and $C_{\mathrm{sal}}$, separated by the star at around t = 1.3 s. Until this point, $C_{\mathrm{sal}}$ is almost constant near the initial value. Accordingly, the transport rate Q evolves in sync with $U_{\mathrm{sal}}$, increasing at the beginning and then showing a plateau. After this point, $C_{\mathrm{sal}}$ rises significantly, showing an overshoot. Alongside the gradual decrease in $U_{\mathrm{sal}}$, the transport rate Q also exhibits a smooth overshoot.

To characterize the moisture effect on the temporal evolution of transport, the time scales for the growth phase of transport rate to reach its peak value ($T_{\mathrm{peak}}$) and the decline phase from the peak moment to the steady state ($T_{\mathrm{decline}}$) are extracted. $T_{\mathrm{peak}}$ is the time duration until the maximum transport rate is reached, while $T_{\mathrm{decline}}$ is the duration from the peak moment until the first moment when the transport rate falls within 15% of the steady-state value. We have checked that the trend of the results is insensitive to the arbitrary 15% criteria for defining $T_{\mathrm{decline}}$. Figure 7a shows that $T_{\mathrm{peak}}$ clearly increases with both the moisture content and the Shields number. In contrast, Figure 7b shows that $T_{\mathrm{decline}}$ decreases first and then increases with the increasing moisture content. This trend of $T_{\mathrm{decline}}$ with the moisture content seems to be inversely related to that of the peak transport rate ($Q_{\mathrm{peak}}$) with the moisture content observed in Figure 7c.

3.2.2. Spatial Evolution

Figure 8 displays the spatial evolution of transport, projected from the temporal results via Equation (26), for all the Shields numbers and moisture contents. As stated in Section “Validation of the Transport Model at the Dry Limit”, under the dry condition, the evolution resembles the solution of a damped harmonic oscillator (see Figure 4a). In contrast, under moist conditions, it presents a more complicated shape, with its peak position shifting towards an increasing distance when the moisture content increases. A plateau is increasingly visible before the peak value is reached when there is more moisture in the bed, consistent with the observations on wet surfaces by [16].

To characterize the moisture effect on the spatial evolution of transport, we analyzed the spatial distances for the transport rate to reach its peak value, $L_{\mathrm{peak}}$. As expected, $L_{\mathrm{peak}}$ increases with both the Shields number and moisture content. Here, we show the trend of $L_{\mathrm{peak}}$ versus the wind shear velocity, $u_{\ast }$, for various moisture contents to relate to other studies [16,61]. Figure 9a suggests a linear correlation between $L_{\mathrm{peak}}$ and $u_{\ast }$ for all the moisture contents. It demonstrates a strong correlation coefficient ($R^2$ = 0.91) and can be expressed using the following equation:

$$L_{\mathrm{peak}}=(A\mathrm{\Omega }+B)(u_{\ast }-u_{\ast ,\mathrm{t}})$$

(28)

$$u_{\ast ,\mathrm{t}}=C\mathrm{\Omega }+D$$

(29)

with A = 205.79 s, B = 4.41 s, C = 0.22 m/s, and D = 0.22 m/s, respectively. $u_{\ast ,\mathrm{t}}$ is the threshold shear velocity. In some field studies, the critical fetch distance is defined as the distance from a non-erodible boundary to where the maximum transport rate occurs [6,16,63,64]. A linear relation between the critical fetch distance and the shear velocity was hypothesized by [61] and adapted by [16] later using field-measured data (hereafter S24; we use “S24” to refer to both the field dataset and the empirical formula from that study). S24 expression is:

$$F_{\mathrm{c}}=100f_{\mathrm{moist}}{\delta }_{\mathrm{sal}}+F_{\mathrm{c},\mathrm{t}}=1.5·{10}^4{f}_{\mathrm{moist}}{\displaystyle \frac{d_{\mathrm{ref}}}{\sqrt{d_{50}}}}{\displaystyle \frac{u_{\ast }-u_{\ast ,\mathrm{t}}}{\sqrt{g}}}+63u_{\ast ,\mathrm{t}}$$

(30)

where $f_{\mathrm{moist}}$ is the moisture effect factor ($f_{\mathrm{moist}}$ = 1 for dry sediment and $f_{\mathrm{moist}}>$1 for moist sand), ${\delta }_{\mathrm{sal}}$ is the saltation height of particles, $d_{\mathrm{ref}}$ is a reference particle diameter with a value of 0.25 mm, and $d_{50}$ is the mean particle diameter. Indicated in the inset of Figure 9a, for dry sediment, $L_{\mathrm{peak}}$ found by our model is clearly lower in value than the critical fetch distance predicted by Equation (30) for some reasons. First, in our simulations, the part of transport from the initiation of grain motion up to the point where our simulation starts is not modeled. Second, in field conditions, the surface is not perfectly dry, as assumed in our model simulations. Beyond these, the grain size, wind condition, and likely other factors may also play additional roles in the fetch distance in field conditions. In the moist scenarios, we found that $L_{\mathrm{peak}}$ shows increasing sensitivity to $u_{\ast }$ at higher moisture contents. This qualitatively matches the observations on wet surfaces by [16], shown in the inset in Figure 9a, and provides insights into the dependency of the critical fetch distance on the surface moisture content.

As stated before, we also run simulations with three different initial conditions, in which different numbers of particles are injected to the system, at $\tilde{\mathrm{\Theta }}$ = 0.06. The results in Figure 9b indicate that the increase in $L_{\mathrm{peak}}$ with moisture content remains consistent across different initial conditions. Furthermore, it can be seen that $L_{\mathrm{peak}}$ is also a function of this initial particle flux under moist conditions, but it remains insensitive to it under dry conditions, matching the observations in [14]. With a higher particle flux fed into the system from upwind, the distance needed to reach the peak value and the steady state decreases. However, further study is needed to establish a comprehensive and detailed relationship between them.

4. Discussion

Our numerical outcomes indicate that steady-state transport rate is insensitive to bed moisture, supporting the claim by [65,66]. This insensitivity is due to the balance between the reduced concentration of saltating particles and their enhanced velocity under moist conditions. The reduced saltation concentration suggests that the ejection of particles is restricted on moist beds, consistent with previous observations [3,7,8,61]. However, the mechanism that leads to the observed increase in the mean saltation velocity under moist conditions still remains unclear. On the one hand, this may be attributed to the reduced number of saltating particles, which causes less deceleration to the wind flow, allowing them to accelerate to higher velocities. On the other hand, it could also be a result of a higher restitution coefficient from the rebounds on moist beds. Studies by [24,65] have shown that rebound dynamics are insensitive to cohesion, lending more support to the first explanation. Accordingly, the average saltation velocity remains almost constant within moist scenarios given an invariant saltation concentration. Even so, it still remains unclear what causes the steady-state saltation concentration to stay constant under moist conditions. To clarify these effects, we will conduct a detailed grain-scale analysis of rebound dynamics, which is the focus of our ongoing work. In addition, experimental studies that systematically measure saltation concentration, velocity, and layer thickness across increasing moisture contents in the steady state of transport are strongly encouraged for model validation.

Our model results support many field observations and empirical relations about transport evolution derived from previous studies. Under dry conditions, the spatial transport evolution projected from the model results is compatible with the solution of a damped harmonic oscillator proposed by [14]. Conversely, the evolution under moist conditions presents a more complicated shape, with an increasingly visible slower growth phase before the actual overshoot as the moisture content increases, qualitatively consistent with observations in the field [16]. Evidenced by previous studies and implied by our simulations, the ejection of new particles is restricted due to the presence of liquid bridges [15,16,27,67], but the mean saltation velocity is enhanced on moist beds. Accordingly, once the ten particles are released, they accumulate momentum from the wind flow gradually by bouncing over the moist surface constantly until their impact velocities reach the threshold for ejecting new particles from the bed. In the most moist case, the initiating particles travel much longer to build up their momentum. Moreover, as evidenced by the higher saltation height, the time required for these particles, once they have gained sufficient momentum, to descend from the air, impact the bed, and eject other particles is also extended. This leads to a more pronounced lag between the peaks of the mean saltation velocity and the saltation concentration. This may explain the slower growth phase of the transport rate before the actual overshoot observed in the transport evolution curves. On the other hand, moisture in the bed has a minimal influence on the particle dynamics in the air, which likely results in the observation that the time scale of the decline phase of transport shows a less clear trend with moisture content. However, the potential inverse relationship between the time scale of the decline phase of the transport rate and its peak value implies that moisture may indirectly influence the decline phase of transport through its impact on peak transport rate. Further investigation is required to confirm this connection.

By projecting the temporal results into a spatial representation, we found that the distance for reaching the peak transport rate, $L_{\mathrm{peak}}$, follows a linear scaling law with the wind shear velocity, consistent with the finding by [14] for dry sediment in wind-tunnel environments and by [16] in field conditions. When compared to the predicted critical fetch distance in field conditions [16], a noticeable difference is observed, the reasons for which have been discussed in Section 3.2.2. Notably, although the initiation of motion of the first grain is not modeled in our model, the distance from the initiation of grain motion up to the point where our model simulation starts could potentially increase with increasing bed moisture content and wind shear condition. For this reason, the found linear relation with a dependency on shear velocity is still plausible. In moist scenarios, our results indicate that this peak distance also increases with moisture content, matching many field observations on moist surfaces [6,11,15,16]. We further found that the dependency of this peak distance on shear velocity and critical shear velocity both present linear relations with moisture content, which puts forward a solid basis for quantifying the moisture effect on transport saturation. Due to the inevitable discrepancy between the simulated transport and real-world situations, which will be discussed later, we highly recommend further investigation on this issue in wind-tunnel and/or field environments.

Furthermore, our simulation results preliminarily suggest that this peak distance is insensitive to the initial particle flux under the dry condition, favoring the finding in the wind-tunnel environment [14]. In contrast, under moist conditions, it is found to decrease with the initial particle flux, which provides important insights into transport on a spatially inhomogeneous surface, e.g., a transition boundary between a dry surface and a moist surface. Extended study is encouraged to quantify the relationship between peak/saturation distance and upwind particle flux under moist conditions.

The periodic boundary conditions used in our model provide a compromising way to simulate an ‘infinite’ domain without having to model the enormous amount of particles, allowing for transport evolution until steady state. Even though compromised, the simulation time for each run still spans weeks. Different from that in the real world, the simulation maintains a uniform and consistent supply of sediment particles compared to a bounded domain where particles could accumulate. The continuous reintroduction of particles maintains a more uniform distribution of momentum across the domain, promoting faster stabilization of transport. Furthermore, in this work, the spatial evolution of transport was translated from the temporal evolution by interpreting the downwind distance as the time integral of the mean saltation velocity. First, the integration ignores the dispersion of individual particles’ velocities. Second, by tracking the modeled frame over the downwind distance, the saltation initiated by wind subsequently in the ‘upwind’ area is excluded. Despite these differences, our numerical results still qualitatively support many previous findings and provide many valuable insights into moisture effects on transport saturation.

In order to focus on the impact of moisture from liquid bridges on the evolution of transport dynamics upon the initiation of transport, evaporation was excluded in the current simulations. However, the estimations in Appendix D suggest that evaporation changes the water content of the sand bed surface and might be substantial under certain environmental conditions. For example, under highly favorable conditions for evaporation—air temperature of 30 °C and relative humidity of 50%—evaporation can deplete the liquid in the topmost particle layer of the sand bed (for the case $\mathrm{\Omega }=1\%$) within seconds to minutes, whereas typical coastal climates lengthen the timescale from tens of seconds to several minutes depending on the resistance to vapor transport. These results justify our omission for the present simulations but also highlight that the resistance parameter for vapor escape may affect the results. However, this is beyond the scope of this study. As mentioned in Section 2.4, wind-driven initiation of transport on moist beds is impossible at Shields numbers corresponding to typical aeolian conditions in our model simulations. This suggests that evaporation from the top sand layer can play a critical role in facilitating an aerodynamic entrainment, consistent with the field observations by [50], and thus should be included in the model for wind-driven initiation of transport. However, as the steady-state transport rate was found to be insensitive to variations in moisture content, it will likely not be affected much by evaporation. Relatedly, evaporation might be further enhanced by the drying of the saltating particles flying through the air after being ejected by particle impact, which might correspond to the dry deposits observed in sand strips, dry bedforms on top of a moist beach surface. However, the dominant effect of bed moisture found on aeolian transport is the inhibition of new grain entrainment by liquid bridges rather than a significant influence on the motion of grains already in saltation.

Other than evaporation, further processes governing moisture variation in the upper sand layers, such as capillary rise from the groundwater head, precipitation, and percolation [68,69], may also play a significant role in aeolian transport. Even at the same overall moisture content, the distribution of water within the sand packing can differ depending on the source. For example, precipitation may lead to a more heterogeneous distribution of liquid bridges compared to saturation from tidal submergence or groundwater fluctuations as rain may selectively wet some grains while leaving others dry. Future work should include these mechanisms, alongside dynamic evaporation, to extend the model’s realism and scope.

5. Conclusions

We presented a novel grain-scale numerical modeling framework, which adds a dynamic viscous force and an evolving migration rule to liquid bridges, for exploring the role of moisture with varying volume contents on the evolution of aeolian sediment transport from a few moving grains towards fully developed steady-state transport. In this work, we isolated the effect of moisture on sediment transport from those mechanisms that might modify the total amount of moisture in the sand bed (e.g., evaporation and groundwater supply). Simulations for a range of wind conditions (Shield numbers $\tilde{\mathrm{\Theta }}$ = 0.01–0.06) and moisture content ($\mathrm{\Omega }$ = 0–20%) revealed that the amount of moisture has little impact on the magnitude of the sand transport rate in the steady-state phase because of its opposing effects on saltation concentration and mean saltation velocity. However, in the transient phase prior to the steady-state transport phase, characterized by the occurrence of a transport overshoot, moisture substantially affects the evolution of transport. It extends the time scale for the growth phase of the transport rate until the peak value of the transport overshoot but shows a less clear influence on the time scale of the transport decline phase. The slower growth phase prior to the onset of transport overshoot for moist cases could be explained by the finding that the sand concentration peaks increasingly later with increasing moisture content compared to the peak in the mean saltation velocity. By projecting the temporal delay in the transport rate overshoot to a spatial equivalent in terms of a downwind shifting of the position of the peak transport rate, the results show that this shift scales linearly with wind shear velocity, with increasing sensitivity to higher moisture content. A preliminary analysis suggests that, in moist conditions, transport saturation is also influenced by the initial particle flux, offering insights into transport behavior on surfaces with varying spatial properties. Moreover, the difficulty of aerodynamic entrainment on moist beds implies that evaporation is essential for modeling the initiation of transport solely by wind, while the steady-state transport rate is expected to remain insensitive to moisture content even if evaporation were included in the model.

This study demonstrates the ability of our grain-scale model to capture key features of transport patterns observed in laboratory and field campaigns, revealing the mechanisms through which moisture influences transport evolution and saturation. These findings help to explain the longer fetch observed on moist beach surfaces and provide a physics-based tool for predicting transport rates under variable moisture levels. While our model currently omits evaporation and groundwater replenishment, integrating these moisture-exchange processes and validating against longer wind-tunnel runs are important subsequent priorities.