3.1. Analytical solutions

For the advection–diffusion equation (2.29) with a sink term, it is known that its solution can be expressed as the product of an exponential decay term and the solution of a corresponding non-stop transport process as given by (2.25) (Zeng & Chen Reference Zeng and Chen2011):

(3.1)\begin{equation} P ( {x,\zeta ,t_0} ) = {P_N} ({x,\zeta ,t_0} )\exp ( { - {k_a}t_0} ). \end{equation}

From a particle-tracking perspective, (3.1) indicates that, while the motion of the particle is governed by a non-stop process (i.e. $P_N$), the probability for this particle to continue its motion after each time step is determined by an exponential function $\exp (-k_a{\rm \Delta} t_0)$ until the particle stops.

Based on the knowledge of Taylor dispersion, we understand that (2.25) cannot be analytically solved for $P_N$, but, instead, that statistical information regarding the bedload particle hops can be obtained through solving the corresponding moment equations (Aris Reference Aris1956). In fact, (3.1) allows us to define the $p$th-order moment of $P(x,\zeta ,t_0)$ as $M_p (\zeta ,t_0)$:

(3.2)\begin{equation} {M_p} ( {\zeta ,t_0} ) = {m_p} ( {\zeta ,t_0} )\exp( { - {k_a}t_0} ), \end{equation}

where

(3.3)\begin{equation} {m_p} ( {\zeta ,t_0} ) = \int _{ - \infty }^{ + \infty } {P_N} ( {x,\zeta ,t_0} ){x^p} \, \textrm{d}x \end{equation}

is the $p$th-order moment for the non-stop bedload transport ($p = 0, 1, 2, \ldots$).

In this work, we consider only the first two statistical moments (i.e. $p = 0$ and 1), which are sufficient for studying the bedload particle hops. As we are going to demonstrate below, the zeroth-order moment $M_0 (\zeta ,t_0)$ is associated with the travel-time distribution of particle hops; and the first-order moment $M_1 (\zeta ,t_0)$ specifies the mean hop distance–travel time relation ($L_h$–$\tau$). Currently, there is only one undetermined parameter ($k_a$), i.e. the deposition rate in (2.29), which can be fitted to the measured data of travel-time p.d.f. when we solve for the zeroth-order moment $M_0 (\zeta ,t_0)$. Although the diffusion coefficient $D$ is already obtained in (2.21) by considering the p.d.f. of the transformed velocity variation ${\rm \Delta} \zeta$, we will demonstrate below how it can be determined alternatively by experimental measurements of travel times and mean hop distances, as a further validation of our theoretical framework.

Applying the operation $\int _{-\infty }^{+\infty }(\,{\cdot }\,){x^p}\,\textrm {{d}}x$ to (2.27)–(2.29), we can obtain the zeroth- and first-order moment equations (i.e. $p = 0$ and 1), respectively, as

(3.4a)\begin{equation} \frac{{\partial {m_0}( {\zeta ,t_0} )}}{{\partial t_0}} = D\frac{{{\partial ^2}{m_0}}}{{\partial {\zeta ^2}}}, \end{equation}

(3.4b)\begin{equation}\frac{{\partial {m_1}( {\zeta ,t_0} )}}{{\partial t_0}} = D\frac{{{\partial ^2}{m_1}}}{{\partial {\zeta ^2}}} - U\log ( {1 - \zeta } ){m_0} ( {\zeta ,t_0} ), \end{equation}

with their initial conditions

(3.5)\begin{equation} {m_p} ( {\zeta ,t_0} ){|_{t_0 = 0}} = \begin{cases} \delta ( {\zeta - {\zeta _0}} ), & {p = 0} ,\\ 0, & {p = 1} , \end{cases} \end{equation}

and corresponding boundary conditions

(3.6)\begin{equation} \left.\frac{{\partial {m_p}}}{{\partial \zeta }}\right|_{\zeta = 0} = \left.\frac{{\partial {m_p}}}{{\partial \zeta }}\right|_{\zeta = 1} = 0. \end{equation}

Considering $\zeta _0\rightarrow 0$ (bedload particles start their motions with a velocity of $u=0$), the zeroth-order moment in (3.4) can be solved as

(3.7)\begin{equation} {m_0} ( {\zeta ,t_0} ) = 1 + 2\sum_{n = 1}^\infty \cos ( {{\beta _n}\zeta } ){\textrm{e}^{ - D\beta _n^2t_0}}, \end{equation}

where $\beta _n=n {\rm \pi},\ n=1, 2, 3, \ldots\,$.

It is obvious that

(3.8a,b)\begin{equation} \langle {{m_0} ( {\zeta ,t_0} )} \rangle = 1 \quad \textrm{and}\quad \langle {{M_0} ( {\zeta ,t_0} )} \rangle = {\textrm{e}^{ - {k_a}t_0}}, \end{equation}

indicating that the number of bedload particles in motion remains the same for non-stop bedload transport, but decays exponentially due to deposition which terminates particle hops. Note that $(1-\langle {{M_0}( {\zeta ,t_0} )} \rangle )$ gives the temporal evolution of the probability that the particle has ceased its motion, which is precisely the definition of the cumulative distribution function (c.d.f.) of the travel time for particle hops. Thus, the corresponding p.d.f. can be analytically determined by differentiating it with respect to the time variable $t_0$:

(3.9)\begin{equation} {f_T} ( t_0 ) = \frac{\textrm{d}}{{\textrm{d}t_0}} ( {1 - \langle {{M_0} ( {\zeta ,t_0} )} \rangle } ) = {k_a}{\textrm{e}^{ - {k_a}t_0}}. \end{equation}

Recall that $t_0$ stands for only half of the travel time (i.e. $t_0 = \tau /2$) based on (2.30). When considering the p.d.f. for the entire travel period, (3.9) should be modified as

(3.10)\begin{equation} {f_T} ( \tau ) = ({k_a}/2 ){\textrm{e}^{ - {k_a}\tau /2}}, \end{equation}

which is an exponential distribution, agreeing with the form observed in experiments (Martin et al. Reference Martin, Jerolmack and Schumer2012; Fathel et al. Reference Fathel, Furbish and Schmeeckle2015; Liu et al. Reference Liu, Pelosi and Guala2019). The mean travel time of the above p.d.f. is $2/k_a$, where the parameter $k_a$ can be determined based on the experimental measurements. For example, Fathel et al. (Reference Fathel, Furbish and Schmeeckle2015) calculated the mean travel time for the particle-tracking experiment as 0.12 s, giving $k_a =16.67$ s$^{-1}$. Equation (3.10) will be used later to determine the analytical solution for the hop-distance distribution.

We note that the physical meaning of the normalized first-order moment (by the proportion of moving particles, here $\langle {{M_0}} \rangle$) is the mean streamwise displacement of bedload particles at a given time, i.e. the mean travel distance of particles with the same half travel time $t_0$. Thus, $\langle {{M_1}} \rangle / \langle {{M_0}} \rangle$ eventually gives the mean hop distance–travel time relation ($L_h$–$\tau$) according to (2.30). Applying the average operator defined in (2.26) on both sides of the first-order moment equation, (3.4b), we obtain

(3.11)\begin{equation} \frac{{\partial \langle {{m_1}} \rangle }}{{\partial t_0}} ={-} U \langle {\log ( {1 - \zeta } ){m_0} ( {\zeta ,t_0} )} \rangle , \end{equation}

which can be solved based on the solution of $m_0$ in (3.7), leading to

(3.12)\begin{align} L(t_0) &= \langle {{M_1}} \rangle / \langle {{M_0}} \rangle = \langle {{m_1}} \rangle / \langle {{m_0}} \rangle \nonumber\\ &=Ut_0 + 2U\sum _{n = 1}^\infty \frac{{1 - {\textrm{e}^{ - D\beta _n^2t_0}}}}{{D\beta _n^3}}\cos ( {{\beta _n}} )\textrm{Si} ( {{\beta _n}} ), \end{align}

where Si$(\,{\cdot }\,)$ is the sine integral function

(3.13)\begin{equation} \textrm{Si} ( {{\beta _n}} ) = \int_0^{{\beta _n}} ({ \sin ( z )}/z ) \, \textrm{d}z. \end{equation}

Again, we note that $t_0$ is only half of the travel time (i.e. $t_0 = \tau /2$); for the entire particle hop, according to (2.30) we have

(3.14)\begin{align} L_h(\tau)&= 2L(t_0)= 2L(\tau/2) \nonumber\\ &=U\tau + 4U\mathop \sum _{n = 1}^\infty \frac{{1 - {\textrm{e}^{ - D\beta _n^2\tau /2}}}}{{D\beta _n^3}}\cos ({{\beta _n}} )\textrm{Si} ( {{\beta _n}} ), \end{align}

which gives the analytical solution for the mean hop distance of an ensemble of particles ($L_h$) as a function of the travel time ($\tau$).

3.2. Mean hop distance–travel time scaling for short and long particle hops

One observation regarding (3.14) is that for long-hop bedload particles (${\tau \to \infty }$), the mean hop distance scales linearly with the travel time:

(3.15)\begin{equation} L_h ( \tau ){|_{\tau \to \infty }} = U\tau + 4U\mathop \sum _{n = 1}^\infty \frac{{\cos ( {{\beta _n}} )\textrm{Si} ( {{\beta _n}} )}}{{D\beta _n^3}}\sim \tau. \end{equation}

This asymptotic regime was discovered recently (Wu et al. Reference Wu, Furbish and Foufoula-Georgiou2020) as a correction to previous studies proposing a single regime for the hop-distance scaling (with an exponent of $\sim 2$ or $\sim 5/3$) of bedload particle motions (Roseberry et al. Reference Roseberry, Schmeeckle and Furbish2012; Fathel et al. Reference Fathel, Furbish and Schmeeckle2015). The scaling regime of $L_h \sim \tau$ was physically explained by resorting to the earlier formulation by Ancey & Heyman (Reference Ancey and Heyman2014) based on the mean-reverting process, which is valid for the description of long-hop particles (Wu et al. Reference Wu, Furbish and Foufoula-Georgiou2020). Under the present formulation invoking Taylor dispersion theory, this linear scaling regime is simply known as the Taylor dispersion regime, when the particle has sampled many times possible velocities within the ‘shear velocity profile’, (2.23). The quote marks represent that it is only mathematically in the same form of the Taylor dispersion process. This understanding indicates that the mean velocity of particle hops converges to a constant $U$ by the time scale $1/D$, where the constant 1 is brought by the normalized velocity $\zeta$. By merely examining the asymptotic regime of the particle hops, i.e. computing the mean velocity of long hops and estimating the starting time of this asymptotic regime, we can roughly estimate the necessary parameters for the governing equation. This point will be further elaborated in later discussion.

As also noted by Wu et al. (Reference Wu, Furbish and Foufoula-Georgiou2020), so far there exists no theory to explain the scaling regime for the short-hop particles, though intuitively it may be attributed to the large proportion of acceleration and deceleration (entrainment and deposition) periods over the entire travel time (of the hop). Owing to the lack of theoretical guidance, and the complicated patterns of short hops, different exponents for the initial scaling regime have been estimated in the literature, and no consensus has been reached for a possibly unified constant. For example, the exponent of 5/3 first appeared in the analysis of a portion of the experimental data by Roseberry et al. (Reference Roseberry, Schmeeckle and Furbish2012) (figure 13 in their paper), most of which are now seen for short hops. With much larger datasets used in a reanalysis, Fathel et al. (Reference Fathel, Furbish and Schmeeckle2015) proposed a new exponent of 2 based on a first estimate of the hop distance–travel time plot (figure 10 in their paper). And this exponent was followed for short hops in the recent study (Wu et al. Reference Wu, Furbish and Foufoula-Georgiou2020) focusing on scaling regime shifts for particle motions.

While (3.14) does not seem to offer an explicit scaling exponent for the short hops, this analytical solution gives us much more detailed and consistent information compared with numerical simulations (Wu et al. Reference Wu, Furbish and Foufoula-Georgiou2020). In figure 7(a) we present results of (3.14) using three different values of the diffusion coefficient $D$, respectively, 100, 10 and 1 s$^{-1}$. The first observation is that different values of $D$ do not affect the long-time asymptotic regime, which agrees with the theoretical expectation of (3.15). However, the time needed for such a transition to the long-time approximation increases as $D$ decreases, which is related to the characteristic time scale $1/D$ by Taylor dispersion theory. Taking the blue curve as an example (for $D = 100$ s$^{-1}$), we can see that one more scaling transition occurred for travel times shorter than $1/D$, as indicated by the two blue solid reference lines with different slopes, which can be used to estimate the scaling exponents as $\sim 1.5$ and $\sim 5/3$, respectively. We can argue that the time scale of $0.1/D$ sets the upper limit of travel times for the initial regime with the scaling exponent of 1.5. This means that particles have a slow mean velocity when starting the hop, which can increase as $\sim \tau ^{0.5}$ if the particle is able to travel longer during a hop. The time scales in between $(0.1/D, 1/D)$ suggest a transition regime. Particle hops with duration falling within the initial and transition regimes can all be defined as short hops, since the mean velocity of these hops has not reached a constant. We note that these scaling regimes could not be identified without the help of the analytical solution of (3.14), given fluctuations and uncertainties observed in numerical and experimental results as shown by figure 4(b) in the paper of Wu et al. (Reference Wu, Furbish and Foufoula-Georgiou2020). The effect of $U$ is more straightforward, as it sets the mean velocity for the long-hop particles (and can thus be determined by the experimental measurements), according to (3.15).

Figure 7.

Scaling regimes for mean hop distances ($L_h$) and travel times ($\tau$) of bedload particle hops. (a) Effects of diffusion coefficient $D$, with two characteristic times of $0.1/D$ and $1/D$ marked for the blue curve of $D = 100$ s$^{-1}$ indicating three stages, respectively, as an initial, a transition and the Taylor dispersion regime. (b) Comparison between the experimental data (Fathel et al. Reference Fathel, Furbish and Schmeeckle2015) for the mean hop distances, which is the same as that presented in figure 4(b) by Wu et al. (Reference Wu, Furbish and Foufoula-Georgiou2020), and the analytical solution (3.14) with parameters of $D = 4$ s$^{-1}$ and $U = 5.56$ cm s$^{-1}$.

The above discussion provides guidelines for estimating the parameters necessary for particle motions in the governing equation (2.29) using measurements of hop distances and travel times. Based on the experimental data (Fathel et al. Reference Fathel, Furbish and Schmeeckle2015) for long hops, we can estimate $U=5.56$ cm s$^{-1}$. It can also be observed from the experimental data that the Taylor dispersion regime for particle hops starts approximately between 0.2 and 0.3 s (Wu et al. Reference Wu, Furbish and Foufoula-Georgiou2020), which gives an estimation of the diffusion coefficient $D$ in the range from 3.33 to 5 s$^{-1}$. Additionally, fitting (3.14) to measured data of the mean hop distances at different travel times, we can determine the diffusion coefficient $D$. Note also that our first estimated value of $D = 4$ s$^{-1}$ in (2.21) based on the p.d.f. of the transformed velocity variation ${\rm \Delta} \zeta$ falls right in the 3.33 to 5 s$^{-1}$ range, corresponding to the average of 0.2 and 0.3 s for $1/D$. The effect of the diffusion coefficient $D$ on the particle motions can be phenomenologically interpreted by considering D as a measure of how fast the particle can change its velocity during the transport. Thus, the faster the particle samples different velocities from the ‘shear velocity profile’ of (2.23) in a Gaussian random walk manner (measured by the standard deviation of the p.d.f. of ${\rm \Delta} \zeta$), the shorter time it needs to experience these velocities many times to approach the Taylor dispersion regime for long hops.

In figure 7(b), we show that the calculated red points for mean hop distances with respect to travel times from the experimental data are perfectly captured by the solid line representing the analytical solution of (3.14). Since the transition regime for the experimental data starts around $0.1/D = 0.025$ s, it can be seen that an exponent larger than 1.5 (here 5/3) explains quite a portion of the scaling behaviour for the measured short hops (mostly in the transition regime), providing a theoretical basis for the estimation of 5/3 made by Roseberry et al. (Reference Roseberry, Schmeeckle and Furbish2012). Note also that the present formulation may explain the estimates of lower exponents (in the range 1.25–1.30) as provided from the experiments by Liu et al. (Reference Liu, Pelosi and Guala2019) in similar transport conditions.

Along with the deposition rate of $k_a =16.67$ s$^{-1}$ as appeared in equation (3.10), all parameters required in the governing equation (2.29) are now determined. We highlight that, based on the diffusion coefficient $D = 4$ s$^{-1}$ and our demarcation on particles performing short hops (i.e. the travel time $\tau < t_s$, with the characteristic time $t_s = 1/D$), for the Fathel et al. (Reference Fathel, Furbish and Schmeeckle2015) experimental dataset, the number of short hops covers over 84 % of the entire hop events according to the travel-time distribution described by (3.10).

3.3. Hop-distance distributions

The hop distance has long been regarded as a key random variable in the formulation of bedload sediment transport, and the form of its distribution is thus of both theoretical and practical importance. Conventionally, the calculation of sediment flux is based on the mean hop distance under equilibrium transport conditions (Einstein Reference Einstein1950), while the second-order moment of the hop-distance distribution comes into play for more generalized cases (Furbish et al. Reference Furbish, Haff, Roseberry and Schmeeckle2012a). Regarding formulations on bedload tracer transport, where an ensemble of tracer particles is followed during the course of experimental observation, the mean of the hop-distance distribution is necessary for the advective term of the tracer transport, whereas the variance contributes to the streamwise diffusion of tracers (Ganti et al. Reference Ganti, Meerschaert, Foufoula-Georgiou, Viparelli and Parker2010; Wu et al. Reference Wu, Foufoula-Georgiou, Parker, Singh, Fu and Wang2019a,Reference Wu, Singh, Fu and Wangb).

The analytical solution of $L_h(\tau )$ for the mean hop distance–travel time relation allows us not only to distinguish between different transport regimes of particle motions depending on how long the particle can travel during a single hop, but also to straightforwardly and theoretically address the specific form of the hop-distance distribution. Based on the work of Fathel et al. (Reference Fathel, Furbish and Schmeeckle2015) (equation (B1) in their paper) and (3.10) for the travel-time distribution $f_T$, we have

(3.16)\begin{equation} {f_L} ( x ) = \left| {\frac{{\textrm{d}{L^{ - 1}}( x )}}{{\textrm{d}x}}} \right| \, {f_T}\left( {{L^{ - 1}} ( x )} \right) = \left| {\frac{{\textrm{d}{L^{ - 1}}( x )}}{{\textrm{d}x}}} \right| \frac{{{k_a}}}{2}\exp\left( { - \frac{{{k_a}}}{2}{L^{ - 1}} ( x )} \right). \end{equation}

In the above equation, $L^{-1} (x)$ is the inverse function of $L_h(\tau )$, which cannot be given explicitly, but can be calculated numerically according to (3.14). In figure 8 we compare the theoretical result of the hop-distance distribution (3.16) with the experimental measurements, based on the obtained parameters $D$, $U$ and $k_a$. The good agreement serves as an excellent validation of the formulation in (2.29) for bedload particle hops. In addition, as also pointed out by Wu et al. (Reference Wu, Furbish and Foufoula-Georgiou2020), the emerging mixed form of the hop-distance distribution, a Weibull-like front and an exponential-like tail, is the result of the different transport regimes for short- and long-hop particles, respectively. That is, the linear scaling of $L_h(\tau ) \sim \tau$ for long hops ensures that the tail of the hop-distance p.d.f. is the same as that of the travel times, which is an exponential distribution. And the scaling of $L_h(\tau ) \sim \tau ^2$ for short hops implies that a different form than the exponential distribution, specifically the Weibull distribution, will emerge for small hop distances in the p.d.f., as concluded by Fathel et al. (Reference Fathel, Furbish and Schmeeckle2015). We note that the tail characteristics of the hop-distance p.d.f. is important in the sense that the heavy-tailed distribution of hop distances was proposed as a possible reason for the observed anomalous diffusion of bedload particle transport (Schumer, Meerschaert & Baeumer Reference Schumer, Meerschaert and Baeumer2009). Thus the thin-tailed distribution of hop distances should indicate the existence of different mechanisms leading to the anomalous diffusion, which, for example, can be related to the heavy-tailed waiting time distribution caused by tracer particles when they stop moving and/or get buried (Liu et al. Reference Liu, Pelosi and Guala2019; Wu et al. Reference Wu, Foufoula-Georgiou, Parker, Singh, Fu and Wang2019a,Reference Wu, Singh, Fu and Wangb).

Figure 8.

The p.d.f. of bedload particle hop distance. (a) Comparison between the experimental data (Fathel et al. Reference Fathel, Furbish and Schmeeckle2015) and the theoretical result of (3.16). The inset shows the same results presented in linear–log plot. (b) Analytical solution for the hop-distance distribution of long hops, in contrast to the Weibull-like front of the distribution for short hops. If there exist only long hops, (3.18) should shift vertically and approach (3.19), which is the exponential p.d.f. for hop-distances conventionally adopted by Einstein (Reference Einstein1950) and others in their research.

In fact, under the current formulation for an exponential distribution of travel times, we can further obtain an explicit expression for the tail of the hop-distance distribution according to the transport regime of the bedload particle performing long hops. Based on the linear relation in (3.15), we have

(3.17)\begin{equation} {L^{ - 1}} ( x ){|_{x \to \infty }}= \frac{1}{U}x - 4\sum _{n = 1}^\infty \frac{{\cos ( {{\beta _n}})\textrm{Si} ( {{\beta _n}} )}}{{D\beta _n^3}}, \end{equation}

which results in

(3.18)\begin{equation} {f_L} ( x ){|_{x \to \infty }}=\frac{{{k_a}}}{{2U}}\exp \left( { - \frac{{{k_a}}}{{2U}}x + 2{k_a}\sum _{n = 1}^\infty \frac{{\cos ( {{\beta _n}} )\textrm{Si}( {{\beta _n}})}}{{D\beta _n^3}}} \right), \end{equation}

indicating an exponential tail of the distribution, agreeing with our qualitative description. According to (3.16), the linear relation of (3.17) simply tells us that the form of the tail of the hop-distance distribution must be the same as that of the travel-time distribution (consistent with observations by Roseberry et al. Reference Roseberry, Schmeeckle and Furbish2012; Liu et al. Reference Liu, Pelosi and Guala2019).

In figure 8(b) we can see that (3.18) captures well the exponential tail of the hop-distance distribution. The Weibull distribution (blue line) in the figure is adopted from Fathel et al. (Reference Fathel, Furbish and Schmeeckle2015): ${f_{LW}}( x ) = \sqrt {{x^{ - 1}}/\langle x\rangle /2} \, \exp ({-\sqrt {2/\langle x\rangle } \, {x^{1/2}}} )$, with the measured mean hop distance $\langle x\rangle = 0.46$ cm. The small differences for short hops between the Weibull distribution and the results of (3.16) are due to different scaling exponents (e.g. 5/3 or 2 for the transition regime) for the mean hop distance–travel time relation. To account for the sharp decay of the probability density by the Weibull-like front due to the large number of particles performing short hops, the black solid line representing (3.18) has to be shifted downwards in the figure as compared with the following exponential p.d.f.:

(3.19)\begin{equation} {f_{Le}} ( x )=\frac{{{k_a}}}{{2U}}\exp \left( { - \frac{{{k_a}}}{{2U}}x} \right). \end{equation}

Note that (3.19) is only for long hops (i.e. hops with a constant mean velocity), and represents the classic form of hop-distance distribution as assumed by Einstein (Reference Einstein1950) and following studies (Paintal Reference Paintal1971; Wu et al. Reference Wu, Foufoula-Georgiou, Parker, Singh, Fu and Wang2019a).

3.4. Validation based on the acceleration distribution

As a further validation of our formulation for the bedload particle hops, we numerically calculated the p.d.f.s of the particle accelerations by particle-tracking simulations based on (2.25). By the algorithm of Gaussian random walk on the $\zeta$-axis (2.24a), and the inverse rule for mapping velocity from $\zeta$-axis to $u$-axis (described by (2.23)), we can generate a time series of velocity $u$ for a particle in motion with the time step of ${\rm \Delta} t = 0.004$ s, which is the same as the interval used for data sampling in experimental measurements. Note that $R$ in (2.24a) is a normally distributed random variable with zero mean and unit standard deviation. The initial and boundary conditions for (2.24a) during the simulations are defined by (2.27) and (2.28), respectively. The same parameter of $D = 4$ s$^{-1}$ is used for the simulation. The accelerations are then obtained by ${\rm \Delta} u /{\rm \Delta} t$, and the results of our theoretical framework agree well with the measurements by Fathel et al. (Reference Fathel, Furbish and Schmeeckle2015) (figure 9). We note that simulated particle accelerations have not been constrained during the formulation of bedload particle hops in this paper (i.e. (2.29)). Hence, the emerging Laplace form of the acceleration p.d.f. is an independent validation and it is also consistent with recent measurements by Liu et al. (Reference Liu, Pelosi and Guala2019).

Figure 9.

Comparison of acceleration p.d.f.s between the experimental data (Fathel et al. Reference Fathel, Furbish and Schmeeckle2015) and results of numerical simulation based on the proposed theoretical framework. The Laplace distribution in the figure (blue line) is $f( {{a_x}} ) = ({1}/{{2\lambda }})\exp ( { - {{| {{a_x}} |}}/{\lambda }} )$, where $a_x$ is the acceleration (cm s$^{-2}$ ) and the parameter $\lambda =610$ cm s$^{-2}$.

We also note that the Laplace-like p.d.f. for the accelerations in figure 9 is not the result of the exponential-like velocity distribution (as we have assumed in (2.14)). We emphasize that, if the particle velocity is randomly drawn each time from the exponential p.d.f. of (2.14) to construct the velocity trajectory as shown by figure 1, the resulting acceleration p.d.f. is indeed Laplace, but with a different variance compared with the results in figure 9. We have performed such numerical simulations to generate the acceleration p.d.f. as demonstrated in figure 10, which is evidently different from the results in figure 9. This can be attributed to the fact that ${\rm \Delta} \zeta$ (difference of nonlinearly transformed velocities) in our model is generated by a normally distributed random variable $R$ as seen in (2.24a), instead of the difference between two velocities randomly drawn from the exponential p.d.f. of (2.14), respectively.

Figure 10.

Demonstration that the acceleration p.d.f. in figure 9 is not the result of the exponential distribution of (2.14). The fitted Laplace distribution (red line) to the simulated accelerations in this figure has a very different variance compared to the Laplace p.d.f. in figure 9 (also reproduced here as the blue line). This is because the numerically simulated accelerations were obtained by differences of two velocities each randomly drawn from the exponential distribution of (2.14), rather than that generated by a normally distributed random variable $R$ as seen in (2.24a) and presented in figure 9.