### CA model

CA models include the coarse graining of sand by considering clusters of sand grains (sand slabs) instead of individual particles. In CA models, the topographic height is taken to be the number of compiled slabs. The topographic height *h(i, j)* at site (*i*, *j*) in a two-dimensional field changes with time. Previous CA models for aeolian sand dunes used a phenomenological formulation of saltation; that is, it was not based on fluid motion. Although there is some variation in the formulations of saltation, the previous models determined situational-specific saltation distances, such as defining the jumping length as a function of the state of the sand bed.

Allen (1982) pointed out that the differences in air and water transport stemmed from differences in the viscosity and density of the medium. On subaqueous beds under relatively slow flows, the main transport mode of sand grains is rolling and sliding, for which bed contact is continuous (Allen 1982; Reid and Frostick 1994; Bridge and Demicco 2008). Therefore, instead of using the saltation length, low transport stages of subaqueous bedforms can be better explained by incorporating changes in the migration speed of grains in continuous contact with the bed. In the simulation used in the present study, the movement of particles was considered separately for two processes: transport due to fluid motion, with continuous bed contact; and avalanching due to gravity. For the remainder of this paper, we will use “bedload” to mean the continuous bed-contact load that is only due to the flow (not due to gravity). Other processes due to flow, such as saltation or suspension, will not be considered so that we can focus on subaqueous environments with low transport rates that form topographies such as sand patches. In the simulation, which process (bedload transport or avalanching) occurs is evaluated deterministically from the geomorphological state of the position. However, the movement of sand slabs due to bedload transport is a stochastic variable that expresses finite velocities for discrete positions; that is, slabs migrate according to a probability that represents the velocity (see below). The simulation ignores the inner structure of the flow, which becomes important when the flow is relatively fast and the bed roughness (bedforms) is large; for simplicity, we ignore the effects of strong erosional vortices and secondary flows.

### Formulation of bedload transport

When simulating bedload transport in a CA model, sand slabs should be restricted to moving to the adjacent downstream cell, unlike the process for saltation. The topographic height is affected by bedload transport as follows:

$$ h\left(i,j\right)\to h\left(i,j\right)-{q}_b\;\mathrm{and}\kern0.5em h\left(i+1,j\right)\to h\left(i+1,j\right)+{q}_b, $$

(1)

where site (*i* + 1, *j*) is downstream of site (*i*, *j*), and *q*
_{
b
} is the number of slabs that migrate during a single calculation step. Here, we consider bedload transport in only the direction of the flow; this is the positive *x*-direction of the two-dimensional (*x*, *y*) system.

If the bed is perfectly uniform, sand may be transported everywhere at a constant rate, but otherwise, the rate of sand migration depends on the location. We formulated a simple phenomenological rule for the probability of sand movement (surrogate of velocity) that does not require calculating the fluid motion. In each time step, a sand slab on the surface moves to the next downstream cell or remains at the same location, as determined stochastically; this probability corresponds to the velocity of the sand migration velocity (that is, a higher probability corresponds to a higher velocity), and this depends on the state of each site.

We made the following assumptions. Sand slabs on a stoss side move fastest at the local peak (state s1), and other sand slabs (for example, partway down the slope) slow down as the downstream gradient increases, due to the competing gravitational force. Sand slabs on lee slopes move more slowly than those on stoss slopes, because to some extent, the bedform shields the downstream flow. For a slab on a lee slope, the probability of movement depends on the extent of the shielding effect of the bedform; we assume that this depends on the gradient of the lee slope. When the slope of a lee side is less than a certain value (state s3), slabs can move as a bedload; otherwise, if the slope is greater than or equal to a threshold value (state s4), slabs are in a perfectly shielded area (shadow zone), in which the flow is not sufficient to convey sand as a bedload, and thus, the sand moves only by avalanching. From these assumptions, we obtain a simple formulation for the state-dependent (s1 to s4) probability, *B*
_{
i, j
}, that the sand at site (*i*, *j*) will migrate by bedload transport to the nearest downstream cell:

$$ {B}_{i,\ j}=\left\{\begin{array}{ll}\alpha \hfill &\ \mathrm{on}\ \mathrm{a}\ \mathrm{l}\mathrm{ocal}\ \mathrm{peak},\ \mathrm{in}\mathrm{cluding}\ \mathrm{a}\ \mathrm{flat}\ \mathrm{top}\ \left(\mathrm{s}1\right)\hfill \\ {}\alpha /\ \left(h\left(i+1,\ j\right)-h\left(i,\ j\right)\right)\hfill & \mathrm{midway}\ \mathrm{a}\mathrm{l}\mathrm{on}\mathrm{g}\ \mathrm{a}\ \mathrm{stoss}\ \mathrm{side}\ \left(\mathrm{s}2\right)\hfill \\ {}\gamma \left\{\beta - \left(h\left(i-1,\ j\right)-h\left(i,\ j\right)\right)\right\}\hfill & \mathrm{on}\ \mathrm{a}\ \mathrm{l}\mathrm{e}\mathrm{e}\ \mathrm{slope},\ \mathrm{but}\ \mathrm{not}\kern0.5em \mathrm{in}\kern0.5em \mathrm{a}\kern0.5em \mathrm{shadow}\ \mathrm{zone}\ \left(\mathrm{s}3\right)\hfill \\ {}0\hfill & \mathrm{in}\ \mathrm{a}\ \mathrm{shadow}\ \mathrm{zone}\ \left(\mathrm{s}4\right),\hfill \end{array}\right. $$

(2)

where *α*, *β*, and *γ* are positive constants. The second expression on the right-hand side shows that the difficulty of climbing a stoss slope depends on the gradient. The third expression on the right-hand side shows the extent to which the bedform reduces the flow, as a function of the gradient of the lee slope.

In Eq. (2), the condition of the local state (s1 to s4) is determined as follows. If *h*
_{
i − 1,j
} ≦ *h*
_{
i,j
}, the site (*i*, *j*) is on a stoss slope or at a local peak. If it is also true that *h*
_{
i + 1,j
} − *h*
_{
i,j
} ≦ 0, then the site is a peak or a flat top (s1); otherwise, it is somewhere along a stoss slope (s2), and the velocity is inversely proportional to the gradient. On the other hand, if *h*
_{
i − 1,j
} > *h*
_{
i,j
}, the site (*i*, *j*) is on a lee slope. If it is also true that *h*
_{
i − 1,j
} − *h*
_{
i,j
} ≦ *β*, then site (*i*, *j*) is not in a shadow zone (s3); otherwise, it is in a shadow zone (s4). Note that *β* is the threshold value for the shadow zone. The above determination of the site state for the bedload (s1 to s4) can be summarized as follows:

$$ \left\{\begin{array}{c}\hfill \mathrm{if}\kern0.5em {h}_{i-1,j}\leqq {h}_{i,j}\kern0.5em \mathrm{and}\left\{\begin{array}{cc}\hfill if\kern0.5em {h}_{i+1,j}-{h}_{i,j}\leqq 0\hfill & \hfill :(s1)\hfill \\ {}\hfill if\kern0.5em {h}_{i+1,j}-{h}_{i,j}>0\hfill & \hfill :(s2)\hfill \end{array}\right.\hfill \\ {}\hfill \mathrm{if}\kern0.5em {h}_{i-1,j}>{h}_{i,j}\kern0.4em \mathrm{and}\left\{\begin{array}{cc}\hfill \mathrm{if}\kern0.5em {h}_{i-1,j} - {h}_{i,j}\leqq \beta \hfill & \hfill :(s3)\hfill \\ {}\hfill \mathrm{if}\kern0.5em {h}_{i-1,j} - {h}_{i,j}>\beta \hfill & \hfill :(s4)\hfill \end{array}\right.\hfill \end{array}\right.. $$

(3)

### Formulation of avalanching

Avalanching occurs as a result of gravity, and consequently, sand can be conveyed by avalanching in a shadow zone, even when bedload transport does not occur. Avalanching occurs when the lee slope is steeper than the angle of repose; the probability that avalanching occurs is unity, which implies deterministic and immediate movement. For avalanching in the streamwise direction at site (*i*, *j*), the topographic height at (*i*, *j*) develops as follows:

$$ h\left(i,j\right)\to h\left(i,j\right)-{q}_a\;\mathrm{and}\kern0.5em h\left(i+1,j\right)\to h\left(i+1,j\right)+{q}_a, $$

(4)

where *q*
_{
a
} is the number of slabs that move by avalanching during one time step. According to flume experiments, avalanching does not occur on a stoss side; thus, we do not need to consider avalanching in the negative *x*-direction. Avalanching always occurs if the following condition is satisfied:

$$ \left\{\mathrm{downward}\ \mathrm{slope}\right\}>{A}_r\ \mathrm{with}\ \mathrm{the}\ \mathrm{condition}(s4), $$

(5)

where *A*
_{
r
} is a positive constant that represents the angle of repose, and the slope in the downstream (positive *x*-direction) is defined as

$$ \left\{\mathrm{downward}\ \mathrm{slope}\right\}=h\left(i, j\right)-h\left(i+1,j\right). $$

(6)

Unlike with bedload transport, because of gravity, avalanching can also occur in the lateral direction:

$$ h\left(i,j\right)\to h\left(i,j\right)-{q}_a\;\mathrm{and}\kern0.5em \left\{h\left(i,j+1\right)\to h\left(i,j+1\right)+{q}_a\;\mathrm{or}\;h\left(i,j-1\right)\to h\left(i,j-1\right)+{q}_a\right\}. $$

(7)

Note that oblique movements between adjacent cells are ignored in the present simulation. Lateral avalanching occurs under the same condition as in Eq. (5), but here, the slopes of interest are those in the positive and negative *y*-directions, which are defined as

$$ \left\{\mathrm{downward}\ \mathrm{slope}\right\}=h\left(i, j\right)-h\left(i,j+1\right),\kern0.5em h\left(i, j\right)-h\left(i,j-1\right), $$

(8)

respectively. If the lateral slopes in both the positive and negative *y*-directions are steeper than the angle of repose, only one of these two directions is selected, and they have equal probability:

$$ {A}_{l+}={A}_{l-}=\frac{1}{2}, $$

(9)

where *A*
_{
l +} and *A*
_{
l −} are the probabilities of avalanching in the positive and negative *y*-direction, respectively. In previous CA models for aeolian dunes, avalanching sand moves in the steepest direction (e.g., Werner 1995). In the present model, for simplicity, we adopted the assumption stated in Eq. (9). When averaged over time, this rule is equivalent to the distribution due to avalanching of sand in a sandpile (e.g., Bak et al. 1987); that is, the sand is distributed equally in all possible directions.

### Simulation

The simulation in the present study was executed as follows. At each time step, the movement of sand slabs in the *x*-direction, due to both bedload transport and avalanching, was determined for each cell and updated in parallel. Next, lateral avalanching (in the *y*-direction) was determined for each cell and updated in parallel. During a single step, avalanching occurred only once per cell (i.e., it was not repeated until perfect relaxation). This was done for simplicity and because we consider bedload transport, and thus, it is not necessary to obtain a static slope (in a strict sense) after perfect relaxation.

In selecting values for the phenomenological parameters, we made the following assumptions. The topographic height was taken to be the number of sand slabs, and therefore, *β* and *A*
_{
r
} were positive integers. Because the slope of a stoss side of a bedform is always gentler than the corresponding lee slope, the inequality 1 < *β* should be satisfied. In addition, we impose the inequality *α* > *γ*(*β* − 1) because the migration of grains is faster on a stoss side than it is on the corresponding lee slope (see Eq. (2) for the gentlest lee slope). The value of *α* is selected arbitrarily, subject to the condition that it be sufficiently smaller than unity; this probability is as a surrogate for the velocity of sand particles. The simulation results are insensitive to *α*, apart from the rate of development for a given time step. For a given value of *α*, the values of *γ* should be sufficiently small that the above inequalities are satisfied; otherwise, the model will not adequately reproduce the shielding effect of lee slopes. For simplicity and to take account of the above inequalities, we used the following values for the parameters: *α* = 0.05, *β* = 2, *γ* = *α*/8(=6.25 × 10^{− 3}), and *A*
_{
r
} = 4. We let *q*
_{
b
} = 1, which represents the assumption that the bedload is a thin layer. We let *q*
_{
a
} = 2 so that there was a fast relaxation of the avalanching; however, we note that when *q*
_{
b
} = 1, the simulation results were very similar, providing *α* was sufficiently smaller than unity. The initial state of the bed was almost flat, with some small random variation, for the available sediment (sand slabs), while the substrate was perfectly flat. The calculation space was a 200 × 40 grid of cells, although we also used a 400 × 80 grid of cells to make particular comparisons. Calculation runs were performed for more than 10^{5} time steps, using a periodic boundary condition. Because the simulation was a minimal model and used the simplest formulae and boundary condition, the main discussion will be restricted to the case of sparse available sediments. We note that when sediments are plentiful, the interference between opposing boundaries cannot be ignored, and thus, it is not appropriate to use a periodic boundary condition.