3.1.1. Cessation threshold

Of the physical parameters affecting the shear velocity at the cessation threshold $u_{\ast t}$, the surface air pressure $P$ varies most strongly with the planetary environment. Furthermore, for a given planetary environment, the grain size $d$ is the most strongly varying relevant physical parameter. To isolate the effect of $P$ on $u_{\ast t}$, we normalise $u_{\ast t}$ in terms of relevant parameters that do neither depend on $P$ nor on $d$, $U_{\ast t}\equiv u_{\ast t}/(\mu g/\rho _p)^{1/3}$ (using that the dynamic viscosity $\mu =\rho _f\nu$ does not depend on $P$), and compare it with the density ratio $s$, which incorporates the effect of $P$ isolated from that of $d$.

For saltation, the simulations reveal a lower bound for $U_{\ast t}$ scaling as $s^{1/3}$ (figure 2, filled circles). This is distinct from the classical scaling of the saltation initiation threshold with $s^{1/2}$ (Greeley et al. Reference Greeley, White, Leach, Iversen and Pollack1976, Reference Greeley, Leach, White, Iversen and Pollack1980; Iversen & White Reference Iversen and White1982; Greeley et al. Reference Greeley, Iversen, Leach, Marshall, White and Williams1984; Burr et al. Reference Burr, Bridges, Marshall, Smith, White and Emery2015, Reference Burr, Sutton, Emery, Nield, Kok, Smith and Bridges2020; Swann et al. Reference Swann, Sherman and Ewing2020) (figure 2, gray crosses), which follows from a balance between flow-induced and resisting forces or torques acting in bed surface grains (Pähtz et al. Reference Pähtz, Clark, Valyrakis and Durán2020). Roughly the same $s^{1/2}$-scaling was also found for the dynamic-threshold measurements by Andreotti et al. (Reference Andreotti, Claudin, Iversen, Merrison and Rasmussen2021) carried out in a low-pressure wind tunnel (figure 2, black crosses). As mentioned in the introduction and discussed in more detail in § 4, these measurements may constitute data of the continuous-transport threshold rather than the cessation threshold.

Figure 2.

Cessation threshold shear velocity normalised using air-pressure- and grain-size-independent natural units $U_{\ast t}\equiv u_{\ast t}/(\mu g/\rho _p)^{1/3}$ versus density ratio $s$. Symbols that appear in the legend correspond to initiation (Greeley et al. Reference Greeley, White, Leach, Iversen and Pollack1976, Reference Greeley, Leach, White, Iversen and Pollack1980; Iversen & White Reference Iversen and White1982; Greeley et al. Reference Greeley, Iversen, Leach, Marshall, White and Williams1984; Burr et al. Reference Burr, Bridges, Marshall, Smith, White and Emery2015, Reference Burr, Sutton, Emery, Nield, Kok, Smith and Bridges2020; Swann, Sherman & Ewing Reference Swann, Sherman and Ewing2020) and cessation threshold measurements for aeolian transport of quartz (Bagnold Reference Bagnold1937; Martin & Kok Reference Martin and Kok2018; Zhu et al. Reference Zhu, Huo, Zhang, Wang, Pähtz, Huang and He2019), clay loam (Chepil Reference Chepil1945) and snow at sea level (Sugiura et al. Reference Sugiura, Nishimura, Maeno and Kimura1998) and high altitude (Clifton, Rüedi & Lehning Reference Clifton, Rüedi and Lehning2006, HA). The dynamic-threshold measurements by Andreotti et al. (Reference Andreotti, Claudin, Iversen, Merrison and Rasmussen2021) may constitute data of the continuous-transport threshold rather than the cessation threshold (discussed in § 4). Symbols that do not appear in the legend correspond to numerical simulations for various combinations of $s$ and the Galileo number $Ga$ (see table 1 and figure 1), with open and filled symbols indicating bedload and saltation conditions, respectively (see figure 1 for the definition). The solid line corresponds to $U_{\ast t}\propto s^{1/3}$ and represents the lower bound for cessation and initiation thresholds of saltation.

In addition to its $s^{1/3}$-scaling, $U_{\ast t}$ varies with the normalised median grain diameter $D_\ast \equiv \sqrt {s}d^+=\sqrt {s}d\tilde {g}/(\tilde {g}\nu )^{2/3}$, described by the following relationship between the rescaled cessation threshold $u_{\ast t}^+$ (note that $u_{\ast t}^+=U_{\ast t}/(s-1)^{1/3}$) and $D_\ast$:

(3.1)\begin{equation} u^+_{{\ast} t}=u^{{+}min}_{{\ast} t}\max\left[\left(\frac{D_\ast}{D^{min}_\ast}\right)^{{-}1/2},\left(\frac{D_\ast}{D^{min}_\ast}\right)^{1/2}\right]. \end{equation}

It contains the parameters $D^{min}_\ast$ and $u^{+min}_{\ast t}$, which denote the location and magnitude, respectively, of the minimum of the function $u_{\ast t}^+(D_\ast )$, corresponding to the lower bound of $U_{\ast t}$ for saltation in figure 2. Equation (3.1) is consistent with the simulations (figure 3a) and experiments (figure 3b) for the saltation regime, though with slightly different parameter values: $(D^{min}_\ast,u^{+min}_{\ast t})=(16,1.6)$ versus $(D^{min}_\ast,u^{+min}_{\ast t})=(18,2.3)$, respectively. The associated relative change of $u_{\ast t}^+$ by $2.3/1.6\simeq 1.4$ is well within the typical systematic uncertainty of cessation threshold measurements. For example, Creyssels et al. (Reference Creyssels, Dupont, Ould El Moctar, Valance, Cantat, Jenkins, Pasini and Rasmussen2009) reported $\varTheta _t=0.009$ for their terrestrial wind tunnel experiments ($d=242\ \mathrm {\mu } \mathrm {m}$), obtained from extrapolating transport rate measurements to vanishing transport using the transport rate model of Ungar & Haff (Reference Ungar and Haff1987), whereas Pähtz & Durán (Reference Pähtz and Durán2020) reported $\varTheta _t=0.0035$ for the very same data using a different transport rate model for the extrapolation, resulting in a relative change of $\sqrt {0.009/0.0035}\simeq 1.6$.

Figure 3.

Rescaled cessation threshold shear velocity $u^+_{\ast t}$ versus normalised median grain diameter $D_\ast \equiv \sqrt {s}d^+$. Symbols in (a) correspond to numerical simulations of saltation (see figure 1 for the definition) for various combinations of the density ratio $s$ and Galileo number $Ga$ (see table 1 and figure 1). Symbols in (b) correspond to experimental cessation threshold data (see legend) for terrestrial aeolian saltation of quartz (Bagnold Reference Bagnold1937; Martin & Kok Reference Martin and Kok2018; Zhu et al. Reference Zhu, Huo, Zhang, Wang, Pähtz, Huang and He2019), clay loam (Chepil Reference Chepil1945) and snow at sea level (Sugiura et al. Reference Sugiura, Nishimura, Maeno and Kimura1998) and high altitude (Clifton et al. Reference Clifton, Rüedi and Lehning2006, HA). The solid lines correspond to (3.1), with $(D_\ast ^{min},u^{+min}_{\ast t})=(16,1.6)$ in (a) and $(D_\ast ^{min},u^{+min}_{\ast t})=(18,2.3)$ in (b).

3.1.2. Equilibrium transport rate

The simulations of saltation and experiments reasonably collapse on the master curve (figure 4)

(3.2)\begin{equation} Q^+{/}d^{{+}3/2}=1.7s^{1/3}(\varTheta-\varTheta_t)+12s^{1/3}(\varTheta-\varTheta_t)^2 \end{equation}

if $Ga\sqrt {s}>81$. The vast majority of planetary transport occurring in nature and most of the simulated saltation conditions satisfy this criterion. Note that $Ga\sqrt {s}$ can be interpreted as a Stokes-like number (Berzi et al. Reference Berzi, Jenkins and Valance2016), encoding the importance of grain inertia relative to viscous drag forcing, and controls the transition to viscous bedload (Pähtz et al. Reference Pähtz, Liu, Xia, Hu, He and Tholen2021).

Figure 4.

Normalised sediment transport rate $s^{-1/3}Q^+/d^{+3/2}$ versus Shields number in excess of the cessation threshold $\varTheta -\varTheta _t$. Symbols in (a) correspond to numerical simulations of saltation (see figure 1 for the definition) for various combinations of the density ratio $s$ and Galileo number $Ga$ (see table 1 and figure 1) with $Ga\sqrt {s}>81$, and Shields number $\varTheta$. Symbols in (b) correspond to measurements for different grain sizes (indicated in the legend) for terrestrial aeolian saltation of minerals (Creyssels et al. Reference Creyssels, Dupont, Ould El Moctar, Valance, Cantat, Jenkins, Pasini and Rasmussen2009; Ho et al. Reference Ho, Valance, Dupont and Ould El Moctar2011; Ho Reference Ho2012; Martin & Kok Reference Martin and Kok2017; Ralaiarisoa et al. Reference Ralaiarisoa, Besnard, Furieri, Dupont, Ould El Moctar, Naaim-Bouvet and Valance2020) and snow (Sugiura et al. Reference Sugiura, Nishimura, Maeno and Kimura1998). The values of $\varTheta _t$ in (b) for a given experimental data set are obtained from extrapolating (3.2) to vanishing transport. Note that Ralaiarisoa et al. (Reference Ralaiarisoa, Besnard, Furieri, Dupont, Ould El Moctar, Naaim-Bouvet and Valance2020) reported that transport may not have been completely in equilibrium in their experiments. The solid lines correspond to (3.2).

3.2.1. First-principle-based proof that $u_{\ast t}^+$ is a function of only $D_\ast$

In general, the shear velocity at the cessation threshold $u_{\ast t}$ is a function of the five control parameters $\rho _p$, $\rho _f$, $\nu$, $\tilde {g}$ and $d$ (Claudin & Andreotti Reference Claudin and Andreotti2006). These parameters involve three units (mass, length and time). According to the $\varPi$ theorem (Barenblatt Reference Barenblatt1996), the physical system, and therefore any dimensionless system property such as $u_{\ast t}^+$, is then controlled by two dimensionless numbers, for example the density ratio $s$ and the normalised median grain diameter $D_\ast$:

(3.3)\begin{equation} u^+_{{\ast} t}=f(s,D_\ast). \end{equation}

To determine the function $f$ in (3.3), existing cessation threshold models have made various idealisations of the fluid–particle system (Claudin & Andreotti Reference Claudin and Andreotti2006; Kok Reference Kok2010b; Berzi et al. Reference Berzi, Jenkins and Valance2016, Reference Berzi, Valance and Jenkins2017; Pähtz & Durán Reference Pähtz and Durán2018a; Andreotti et al. Reference Andreotti, Claudin, Iversen, Merrison and Rasmussen2021; Pähtz et al. Reference Pähtz, Liu, Xia, Hu, He and Tholen2021; Gunn & Jerolmack Reference Gunn and Jerolmack2022). In particular, they all drastically coarse-grain the particle phase of the aeolian transport layer above the bed surface, either by representing the entire grain motion by identical periodic saltation trajectories (Claudin & Andreotti Reference Claudin and Andreotti2006; Kok Reference Kok2010b; Berzi et al. Reference Berzi, Jenkins and Valance2016, Reference Berzi, Valance and Jenkins2017; Pähtz & Durán Reference Pähtz and Durán2018a; Andreotti et al. Reference Andreotti, Claudin, Iversen, Merrison and Rasmussen2021; Pähtz et al. Reference Pähtz, Liu, Xia, Hu, He and Tholen2021; Gunn & Jerolmack Reference Gunn and Jerolmack2022) or by an average motion behaviour (Kok Reference Kok2010b; Pähtz & Durán Reference Pähtz and Durán2018a) that is mathematically equivalent to an identical periodic trajectory representation (Pähtz et al. Reference Pähtz, Clark, Valyrakis and Durán2020).

Here, in contrast to previous models, we do not resort to any such coarse-graining. Instead, we idealise the system in the following comparably mild manner.

1. (i) We consider only buoyancy and Stokes drag as fluid–grain interactions, neglecting form drag contributions. This would be justified if relatively fast saltating grains dominated the near-threshold grain dynamics, since comparably faster grains exhibit comparably lower fluid–particle velocity differences and, thus, comparably less form drag relative to Stokes drag.

2. (ii) Due to the typically relatively small shear Reynolds numbers associated with planetary transport near the cessation threshold, $Ga\sqrt {\varTheta _t}\lesssim 10$, we consider a smooth inner turbulent boundary layer mean flow velocity profile $u_x(z)$, neglecting hydrodynamically rough contributions (and turbulent fluctuations, which are also neglected in the numerical simulations).

3. (iii) Since vanishingly few grains are in motion sufficiently close to the cessation threshold, we neglect the feedback of the grain motion on the flow.

4. (iv) Since saltation trajectories are typically much larger than the grain size, we consider an idealised flat bed and assume that the zero level of the flow velocity coincides with the grain elevation at grain–bed impact ($z=0$), neglecting the effect of the flow very near the bed surface to the overall grain motion.

5. (v) While we do not specify the distribution of grain lift-off velocities $f_\uparrow$ and grain impact velocities $f_\downarrow$, we assume that the boundary conditions mapping $f_\downarrow$ to $f_\uparrow$ in the steady state are scale-free, as for grain–bed rebounds (Beladjine et al. Reference Beladjine, Ammi, Oger and Valance2007), neglecting the potential effect of $\sqrt {\tilde {g}d}$ on grain–bed collisions. Most grains ejected by the splash of a grain impacting the bed with velocity $\boldsymbol {v}_\downarrow$ exhibit a velocity on the order of $\sqrt {\tilde {g}d}$ and only the few grains corresponding to the upper-tail end of the distribution exhibit an ejection velocity proportional to $|\boldsymbol {v}_\downarrow |$ (Lämmel et al. Reference Lämmel, Dzikowski, Kroy, Oger and Valance2017). Hence, this assumption effectively means that grain–bed rebounds and/or rare extreme ejection events dominate the saltation dynamics relevant for the cessation threshold scaling.

Under the above assumptions, the equations of motion for a given grain are (Pähtz et al. Reference Pähtz, Liu, Xia, Hu, He and Tholen2021)

(3.4)\begin{gather} \dot{v}^+_z={-}1-v^+_z/v^+_s, \end{gather}

(3.5)\begin{gather}\dot{v}^+_x=(u^+_x-v^+_x)/v^+_s, \end{gather}

(3.6)\begin{gather}u^+_x=u^+_\ast f_u(u^+_\ast z^+), \end{gather}

where $\boldsymbol {v}^+$ is the rescaled grain velocity, $v_s^+=4sd^{+2}/(3Re_c)$ the rescaled Stokes settling velocity (obtained from the high-viscosity limit of (2.1)), and $f_u(X)$ denotes a function describing $u_x/u_\ast$ for an undisturbed smooth inner turbulent boundary layer. It obeys $f_u(X)=X$ within the viscous sublayer of the turbulent boundary layer ($X\lesssim 5$) and $f_u(X)\simeq \kappa ^{-1}\ln (9X)$ within its log-layer ($X\gtrsim 30$). Extrapolated into the transitional buffer layer in between, both profiles would intersect at about $X=11$, which is why $\delta _\nu =11\nu /u_\ast$ is termed viscous-sublayer thickness.

Parametrised by $v^+_s$ and $u^+_\ast$, (3.4)–(3.6) map $f_\uparrow$ to $f_\downarrow$. Combined with the scale-free boundary conditions, mapping $f_\downarrow$ back to $f_\uparrow$, they imply that the grain motion is fully determined by $v^+_s$ and $u^+_\ast$. For a given $v^+_s$, the cessation threshold $u^+_{\ast t}$ then corresponds to the smallest value of $u^+_\ast$ for which a solution of the combined system exists (Pähtz et al. Reference Pähtz, Liu, Xia, Hu, He and Tholen2021). This implies that there is a function $f$ mapping $D_\ast =\sqrt {18v^+_s}$ (valid for $Re_c=24$, the standard case of non-rarefied drag) to $u^+_{\ast t}$:

(3.7)\begin{equation} u^+_{{\ast} t}=f(D_\ast). \end{equation}

In summary, the above assumptions simplify the general two-parametric dependence of $u_{\ast t}^+$ in (3.3) to the one-parametric dependence in (3.7), in agreement with (3.1).

3.2.2. Semi-empirical model of cessation threshold scaling

While the above analysis explains why $u_{\ast t}^+=f(D_\ast )$ in (3.1), it does not yield the function $f$ itself. Here, we derive the expression for $f$ in (3.1) guided by the simulations. The latter show that the minimum $u^{+min}_{\ast t}$ for saltation occurs when the hop height $\overline {v_z^2}_t/\tilde {g}$ is about equal to the viscous-sublayer thickness $\delta _{\nu t}=11\nu /u_{\ast t}$ near the cessation threshold (figure 5a). This can be explained using the empirical, yet physically reasonable, simulation-supported proportionality between the average fluid velocity $\overline {u_x}_t$ and $(\overline {v_z^2})^{1/2}_t$ near the cessation threshold (figure 5b). In fact, averaging (3.6) over all grain trajectories and the transport layer, using the approximation $\overline {f_u(u^+_{\ast t}z^+)}_t\simeq f_u(u^+_{\ast t}\overline {z^+}_t)$, and using this proportionality approximately yields for saltation ($\overline {z^+}_t\simeq \overline {v_z^{+2}}_t$, see figure 1):

(3.8)\begin{equation} u^+_{{\ast} t}\propto\left[\frac{u^+_{{\ast} t}\overline{v_z^{{+}2}}_t}{f_u^2\left(u^+_{{\ast} t}\overline{v_z^{{+}2}}_t\right)}\right]^{1/3}. \end{equation}

Within the viscous sublayer ($u^+_{\ast t}\overline {v_z^{+2}}_t\lesssim 5$), this relation simplifies to $u^+_{\ast t}\propto (u^+_{\ast t}\overline {v_z^{+2}}_t)^{-1/3}$ and within the log-layer approximately to $u^+_{\ast t}\propto (u^+_{\ast t}\overline {v_z^{+2}}_t)^{1/3}$, neglecting the logarithmic term. The crossover between the two power laws occurs about at $u^+_{\ast t}\overline {v_z^{+2}}_t=11$, that is, when the hop height exceeds the viscous-sublayer thickness ($\overline {v_z^2}_t/\tilde {g}=\delta _{\nu t}$). Hence, the parabolic law

(3.9)\begin{equation} u^+_{{\ast} t}=u^{{+}min}_{{\ast} t}\max\left[\left(\frac{\overline{v_z^2}_t}{\tilde{g}\delta_{\nu t}}\right)^{{-}1/3},\left(\frac{\overline{v_z^2}_t}{\tilde{g}\delta_{\nu t}}\right)^{1/3}\right] \end{equation}

fits the saltation data reasonably well (solid line in figure 5a).

Figure 5.

(a) Rescaled threshold shear velocity $u^+_{\ast t}$ versus ratio between hop height $\overline {v_z^2}_t/\tilde {g}$ and viscous-sublayer thickness $\delta _{\nu t}=11\nu /u_{\ast t}$ near the cessation threshold. (b) Rescaled transport layer-averaged fluid velocity $\overline {u^+_x}_t$ versus $(\overline {v_z^{+2}})^{1/2}_t$ near the cessation threshold. (c) Plot of $u^+_{\ast t}\overline {v_z^{+2}}_t$ versus $v^+_s$. Symbols correspond to numerical simulations of saltation (see figure 1 for the definition) for various combinations of the density ratio $s$ and Galileo number $Ga$ (table 1 and figure 1). The solid lines in (a), (c) and (b) correspond to (3.9), (3.10) and $\overline {u^+_x}_t=6(\overline {v_z^{+2}})^{1/2}_t$, respectively.

Following from the analysis we have used to deduce (3.7), the grain kinematics near the cessation threshold, and thus $\overline {v_z^{+2}}_t$, should be controlled by $u^+_{\ast t}$ or $v^+_s$. Indeed, the simulations of saltation suggest the empirical relation (figure 5c)

(3.10)\begin{equation} u^+_{{\ast} t}\overline{v_z^{{+}2}}_t=1.5v_s^{{+}3/4}, \end{equation}

which leads to (3.1) with $D_\ast ^{min}=\sqrt {18}(11/1.5)^{2/3}\simeq 16$.

According to the above model, the grain size scaling of $u^+_{\ast t}$ in (3.1), despite being mathematically equivalent to the well-known cohesive ($u_{\ast t}\sim d^{-1/2}$, left branch) and cohesionless ($u_{\ast t}\sim d^{1/2}$, right branch) limits of the saltation initiation threshold (Shao & Lu Reference Shao and Lu2000), follows purely from hydrodynamics rather than the onset of cohesion at small grain sizes.

3.2.3. Physics behind equilibrium transport rate scaling

Analytical, physical models of the equilibrium transport rate $Q$ for aeolian saltation typically separate it into the mass of transported sediment per unit area of the bed $M$ and its average streamwise velocity $V$ through $Q=MV$. In most models, it is reasoned that the scaling of $V$ is in one way or another linked to grain–bed collisions, and since the average outcome of grain–bed collisions should be roughly independent of the wind speed at equilibrium, $V$ is taken as equal to its near-threshold value $V_t$ (Ungar & Haff Reference Ungar and Haff1987; Durán et al. Reference Durán, Claudin and Andreotti2011; Kok et al. Reference Kok, Parteli, Michaels and Karam2012; Berzi et al. Reference Berzi, Jenkins and Valance2016). However, it has been shown that, for sufficiently intense saltation, midair collisions significantly disturb grain trajectories (Carneiro et al. Reference Carneiro, Araújo, Pähtz and Herrmann2013; Pähtz & Durán Reference Pähtz and Durán2020; Ralaiarisoa et al. Reference Ralaiarisoa, Besnard, Furieri, Dupont, Ould El Moctar, Naaim-Bouvet and Valance2020), leading to an additional additive term increasing as $M^+/d^+$ (Pähtz & Durán Reference Pähtz and Durán2020):

(3.11)\begin{equation} Q^+=M^+V^+_t(1+c_MM^+{/}d^+), \end{equation}

where $c_M$ is a constant parameter. It is not trivial to evaluate the scalings of $M^+$ and $V^+_t$ with the simulation data, since extracting $M$ and $V$ from DEM-based numerical transport simulations is ambiguous (Durán et al. Reference Durán, Andreotti and Claudin2012; Pähtz & Durán Reference Pähtz and Durán2018b). One possible way is to define $M$ as the mass $M_0$ of grains moving above the bed surface ($z=0$) per unit bed area and $V$ as their average streamwise velocity (Pähtz & Durán Reference Pähtz and Durán2018b):

(3.12)\begin{gather} M\equiv\int_0^\infty\rho\,\mathrm{d} z=M_0, \end{gather}

(3.13)\begin{gather}V\equiv\frac{\displaystyle\int\nolimits_0^\infty\rho\langle v_x\rangle\,\mathrm{d} z}{ \displaystyle\int\nolimits_0^\infty\rho\,\mathrm{d} z}=\overline{v_x}. \end{gather}

This definition uses that most (but not all) sediment transport occurs at elevations $z>0$, especially for saltation and, therefore, $M_0\overline {v_x}=\int _0^\infty \rho \langle v_x\rangle \,\mathrm {d} z\simeq \int _{-\infty }^\infty \rho \langle v_x\rangle \,\mathrm {d} z=Q$ (Pähtz & Durán Reference Pähtz and Durán2018b). Alternatively, one can define $V$ as the mass flux-weighted average $\overline {v_x}^q$ of the streamwise velocity of all grains and $M_q$, the associated value of $M$, as $M_q\equiv Q/\overline {v_x}^q$ (Durán et al. Reference Durán, Andreotti and Claudin2012):

(3.14)\begin{gather} M\equiv\frac{\left(\displaystyle\int\nolimits_{-\infty}^\infty\rho\langle v_x\rangle\,\mathrm{d} z\right)^2}{ \displaystyle\int\nolimits_{-\infty}^\infty\rho\langle v_x^2\rangle\,\mathrm{d} z}=M_q, \end{gather}

(3.15)\begin{gather}V\equiv\frac{\displaystyle\int\nolimits_{-\infty}^\infty\rho\langle v_x^2\rangle\,\mathrm{d} z}{ \displaystyle\int\nolimits_{-\infty}^\infty\rho\langle v_x\rangle\,\mathrm{d} z}=\overline{v_x}^q, \end{gather}

where $\bar {{\cdot }}^q\equiv ({1}/{Q})\int _{-\infty }^\infty \rho \langle v_x{\cdot }\rangle \,\mathrm {d} z$.

For the above two definitions of $M$ and $V$, the simulations are roughly described by scaling laws in which a comparably small part of the $s^{1/3}$-scaling factor in (3.2) goes into $M^+/d^+$ and a comparably large part into $V^+_t/\sqrt {d^+}$ (figure 6). However, the exact partitioning of $s^{1/3}$ depends on the chosen definition (figures 6(a) and 6(b) versus figures 6(c) and 6(d)):

(3.16a,b)\begin{gather} M^+_0\propto s^{1/12}d^+(\varTheta-\varTheta_t),\quad\overline{v^+_x}_t\propto s^{1/4}\sqrt{d^+}, \end{gather}

(3.17a,b)\begin{gather}M^+_q\propto d^+(\varTheta-\varTheta_t),\quad\overline{v^+_x}^q_t\propto s^{1/3}\sqrt{d^+}. \end{gather}

The latter scaling is consistent with the prediction $M^+\propto d^+(\varTheta -\varTheta _t)$ from physical models (Ungar & Haff Reference Ungar and Haff1987; Durán et al. Reference Durán, Claudin and Andreotti2011; Berzi et al. Reference Berzi, Jenkins and Valance2016; Pähtz & Durán Reference Pähtz and Durán2020) and with (3.2) when combined with (3.11). However, it means that $V^+_t\propto s^{1/3}\sqrt {d^+}$, which is a highly unusual scaling, different from the existing models $V^+_t\propto \sqrt {d^+}$ (Ungar & Haff Reference Ungar and Haff1987; Berzi et al. Reference Berzi, Jenkins and Valance2016) and $V^+_t\propto u^+_{\ast t}$ (Durán et al. Reference Durán, Claudin and Andreotti2011; Kok et al. Reference Kok, Parteli, Michaels and Karam2012; Pähtz & Durán Reference Pähtz and Durán2020).

Figure 6.

(a,c) Normalised transport loads $s^{-1/12}M^+_0/d^+$ and $M^+_q/d^+$, using the definitions (3.12) and (3.14), respectively, of $M$; and (b,d) normalised average streamwise grain velocities $s^{-1/4}\overline {v^+_x}/\sqrt {d^+}$ and $s^{-1/3}\overline {v^+_x}^q/\sqrt {d^+}$, using the definition (3.13) and (3.15), respectively, of $V$ versus Shields number in excess of the cessation threshold $\varTheta -\varTheta _t$. Symbols correspond to numerical simulations of saltation (see figure 1 for the definition) for various combinations of the density ratio $s$ and Galileo number $Ga$ (see table 1 and figure 1) with $Ga\sqrt {s}>81$, and Shields number $\varTheta$. The solid lines in (a,b) correspond to the left equations in (3.16a,b) and (3.17a,b), respectively.

3.3.1. Test of cessation threshold models

The most important assumption that led to the simulation-supported statement that the rescaled cessation threshold $u_{\ast t}^+$ is solely controlled by the normalised median grain diameter $D_\ast$ in § 3.2.1 is that of scale-free boundary conditions. The only existing cessation threshold model with scale-free boundary condition is that of Pähtz et al. (Reference Pähtz, Liu, Xia, Hu, He and Tholen2021), which we here compare with the most recent alternative, that of Gunn & Jerolmack (Reference Gunn and Jerolmack2022). The latter's most important feature is that it superimposes a $Ga$-dependent damping on the scale-free laws describing grain–bed rebounds, where the damping function is essentially fitted to agreement with experimental cessation threshold data. We find that, while the model of Pähtz et al. (Reference Pähtz, Liu, Xia, Hu, He and Tholen2021) captures the simulation data very well, the model of Gunn & Jerolmack (Reference Gunn and Jerolmack2022), with its drag and lift laws being modified to those employed in the simulations (i.e. (2.1) and no lift) for a fair comparison, is in very strong disagreement (figure 7). This is discussed in § 4.

Figure 7.

Evaluation of the cessation threshold models of (a) Pähtz et al. (Reference Pähtz, Liu, Xia, Hu, He and Tholen2021) and (b) Gunn & Jerolmack (Reference Gunn and Jerolmack2022), where the latter's drag and lift laws have been modified to those employed in the simulations for a fair comparison. Rescaled cessation threshold $u_{\ast t}^+$ versus normalised median grain diameter $D_\ast \equiv \sqrt {s}d^+$. Symbols correspond to numerical simulations of saltation (see figure 1 for the definition) for various combinations of the density ratio $s$ and Galileo number $Ga$ (see table 1 and figure 1). The solid lines indicate the respective model predictions. Their color characterises $s$ in accordance with the symbol color.

3.3.2. Test of equilibrium transport rate models

The simulations of saltation are not or not well captured by the two most widely used physical models of the equilibrium aeolian transport rate: the model of Ungar & Haff (Reference Ungar and Haff1987) and others (e.g. Jenkins & Valance Reference Jenkins and Valance2014; Berzi et al. Reference Berzi, Jenkins and Valance2016), $Q^+/d^{+3/2}=f_1(\varTheta -\varTheta _t)$ (figure 8a) and the model of Durán et al. (Reference Durán, Claudin and Andreotti2011) and others (Kok et al. Reference Kok, Parteli, Michaels and Karam2012; Pähtz & Durán Reference Pähtz and Durán2020), $Q^+/(d^+u^+_{\ast t})=f_2(\varTheta -\varTheta _t)$ (figure 8b).

Figure 8.

Evaluation of the physically based functional relationships for the sediment transport rate by Ungar & Haff (Reference Ungar and Haff1987) and Durán et al. (Reference Durán, Claudin and Andreotti2011). Normalised sediment transport rate (a) $Q^+/d^{+3/2}$ and (b) $Q^+/(d^+u^+_{\ast t})$ versus Shields number in excess of the cessation threshold $\varTheta -\varTheta _t$. Symbols correspond to numerical simulations of saltation (see figure 1 for the definition) for various combinations of the density ratio $s$ and Galileo number $Ga$ (see table 1 and figure 1) with $Ga\sqrt {s}>81$, and Shields number $\varTheta$.