Linear-stability analysis of plane beds under flows with suspended loads

The coefficient bn is derived as

b = K−1 NR1 .

(A28)

Therefore, the following equation is obtained:

c1 (η) = c1∗ (η)R1 .

(A29)

By substituting Eqs. (A17), (A18), and (A29) into Exner’s

equation (Eq. 61), the complex angular frequency ω is obtained in the following form:

ω = ω(k, F r, Cz , Rep ) = ωr + iωi ,

(A30)

where ωi corresponds to the growth rate of the perturbation.

˜ h˜ 0 ) (Eq. 27) and

Here, using Rep = Rep (D) = Rep (D,

˜ h˜ 0 ) (Eq. 52), we can rewrite Eq. (A30)

Cz = Cz (R0 ) = Cz (D,

as

˜ h˜ 0 .

ω = ω k, F r, D,

(A31)

Thus, we can obtain the growth rate ωi as a function of k

˜ and h˜ 0 .

for a given combination of Fr, D,

Code and data availability. The datasets and codes used for

this study can be found at https://doi.org/10.5281/zenodo.8332448

(Ohata et al., 2023). Unpublished data used for the analysis were cited from the dataset of Brownlie (2018)

(https://doi.org/10.22002/d1.943).

Author contributions. KO and NI performed the linear-stability

analysis. HN and NI contributed to the interpretation of the results.

KO wrote the manuscript and prepared the figures, and then HN and

NI provided feedback on the manuscript and figures.

Competing interests. The contact author has declared that none

of the authors has any competing interests.

Disclaimer. Publisher’s note: Copernicus Publications remains

neutral with regard to jurisdictional claims in published maps and

institutional affiliations.

Acknowledgements. We would like to express our gratitude to

Robert Dorrell for his comments. We are thankful to the anonymous referees for their insightful comments on earlier versions of

the manuscript.

https://doi.org/10.5194/esurf-11-961-2023

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