Well-Posedness and Stability Analysis of a Landscape Evolution Model

Well-Posedness and S abili y Analysis o a Landscape E olu ion Model Julie Bina d1 · Pie e Degond2 · Pascal Noble1 Recei ed: 17 No embe 2022 / Accep ed: 4 No embe 2023 / Abs ac In his pape , we s udy a sys em o pa ial di e en ial equa ions modeling he e olu ion o a landscape in o de o desc ibe he mechanisms o pa e n o ma ions. A g ound su ace is e oded by he low o wa e o e i , by ei he sedimen a ion o dilu ion. We conside a model, composed o h ee e olu ion equa ions: one on he ele a ion o he g ound su ace, one on he luid heigh and one on he concen a ion o sedimen s in he luid laye . We i s es ablish he well-posedness o he sys em in sho ime and unde he assump ion ha he ini ial luid heigh does no anish. Then, we ocus on pa e n o ma ion in he case o a ilm low o e an inclined e odible plane. Fo ha pu pose, we ca y ou a spec al s abili y analysis o cons an s a e solu ions in o de o de e mine ins abili y condi ions and iden i y a mechanism o pa e n o ma ions. These pa e ns, which a e ills and gullies, a e he s a ing poin o he o ma ion o i e s and alleys in landscapes. Finally, we ca y ou some nume ical simula ions o he ull sys em in o de o alida e he spec al ins abili y scena io, and de e mine he esul ing pa e ns. Keywo ds Pa e n o ma ion · Ins abili ies · Landscape e olu ion model · E osion by wa e · S eam incision law · Sedimen a ion Communica ed by Jose A. Ca illo. BJulie Bina d [email p o ec ed] Pie e Degond [email p o ec ed] - oulouse. Pascal Noble [email p o ec ed] 1Ins i u de Ma héma iques de Toulouse, UMR 5219, CNRS, INSA, Uni e si é de Toulouse, 31077 Toulouse, F ance 2Ins i u de Ma héma iques de Toulouse, UMR 5219, CNRS UPS, Uni e si é de Toulouse, 31062 Toulouse Cedex 9, F ance 20 Page 2 o 55 Ma hema ics Subjec Classi ica ion 35M30 ·35Q86 ·35B35 ·35B36 ·35A01 ·86-10 1 In oduc ion The modeling o landscape e olu ion unde he e ec o wa e low has ecei ed inc easing a en ion hese las decades. Se e al physical phenomena mus be consid- e ed in o de o desc ibe he e osion and sedimen a ion p ocesses, which occu when wa e lows o e an e odible su ace. The e osion is he emo al o sedimen s om he soil by a luid, and he sedimen a ion is he in e se p ocess, when he sedimen s se le on he su ace. Heu is ically, he in ensi y o he e osion p ocess is s ongly ela ed o he low a e o he luid, he e osion a e is highe when he low a e is highe . An o he p ocess ha a ec s landscape e olu ion is he c eep e ec , which ends o smoo h he bo om su ace on imescales ha a e much la ge han he ones in ol ed in sedimen anspo . This e ec has mul iple causes, such as g a i a ional o ces ac ing on he soil, and co osion and dila a ion caused by physical and chemical ac o s. I can be desc ibed as a simple di usion p ocess o he soil. The c eep e ec was in oduced in 1892, in Da is (1892). I was used in Gilbe (1909) o explain he con exi y o hill ops p o ile. La e , in 1963 a de i a ion o he soil c eep e ec as a limi o a s ochas ic p ocess had been done in Culling (1963). The s ochas ic e ec models soil pa icles which ollow a andom mo ion, unde he cons ain o g a i y. In his a icle, Culling w i es “Soil c eep—so g adual as o be impe cep ible—appea s o be he esul o he pe sis en e ec o molecula and mac omolecula o ces ending o displace he soil pa icles,” and he desc ibes he soil low as a quasi- iscous luid. The desc ip ion o he geological p ocesses ha occu in landscape e olu ion was ini ia ed in Gilbe (1877). In his book, he au ho gi es he undamen al p inciples o landscape e olu ion and explains why he p o iles o s eam beds a e conca e upwa ds. This conca i y p ope y had been illus a ed la e in Culling (1960), wi h a ma hema ical desc ip ion o he e osion o a slope. The slope e olu ion is desc ibed wi h a eac ion di usion equa ion, he exponen ial sou ce e m ep esen ing he e osion a e. The landscapes e ol e mainly in unc ion o hese wo compe i i e ac o s: c eep on he hill ops and s eam incision on he lowe slopes. Indeed, in he uppe slopes he wa e low is weak and dispe si e; hus, he c eep e ec p edomina es. Downwa d, he e ec o shea s ess o he low becomes dominan , and bedload anspo p ocess inc eases. This leads o he o ma ion o s eam beds and alley, he p o iles being conca e. Since hen, he complexi y o he landscape e olu ion models inc eased: See Chen e al. (2014a,b) o a e iew o hese models. The p ocess o e osion and anspo o sedimen s can be desc ibed by wo di e en laws, he allu ial anspo law o he s eam incision law, which depends on he na u e o he soil. When he soil is co e ed by allu iums, he sedimen s a e di ec ly anspo ed by wa e lux, and he amoun o sedimen s mo ed by he wa e lux in a gi en ime qs ollows a law ha depends on he wa e discha ge q. This anspo discha ge law is desc ibed, o example, in Smi h and B e he on (1972) and is gi en by qs=kqm|∇z|n(1) Page 3 o 55 20 whe e k,mand na e cons an s. This is called he anspo limi ed case. On he o he side, he s eam incision law is used when he su ace is bed ock, because in his case he sedimen anspo is limi ed by he esis ance o he bed ock o he shea s ess caused by he wa e lux. This is modeled by an e osion sou ce e m in he equa ion o su ace heigh e olu ion, as desc ibed in Howa d and Ke by (1983). The sedimen s emo ed om he soil by he e osion a e supposed o be dissol ed in wa e and hus a e anspo ed by he wa e lux. This s eam incision law is desc ibed in Sec .2. This case ha we s udy in his pape is called he de achmen limi ed case. In his pape , we s udy a sys em o pa ial di e en ial equa ions desc ibing he e osion o he soil by wa e , which had been p oposed in Chen e al. (2014a), Sec ion 4. I includes he modeling o he wa e low, he e osion o he su ace by wa e , he anspo and deposi ion o sedimen s, and he c eep e ec . The cha ac e is ic luid eloci y is usually much la ge han he e osion a e: In o de o desc ibe pa e n o ma ion, we only conside la ge-scale luc ua ions o he luid eloci y. As a esul , we assume ha he luid eloci y is p opo ional o he g adien o he ee su ace ele a ion. The o he physical p inciples aken in accoun a e: •The conse a ion law o wa e and sedimen s dissol ed in wa e , •The s eam incision law: he e osion g ows wi h he wa e speed and he wa e heigh , •The sedimen a ion a e is p opo ional o he concen a ion o sedimen s in wa e , •The c eep e ec : he soil is subjec o a di usion p ocess. O he e ec s, such as in il a ion, ege a ion, wind o ice o ma ion, a e neglec ed. Mo eo e , we suppose ha he bo om su ace is cons i u ed by one ype o sedimen s. The sys em s udied in his a icle models he ime e olu ion o he soil and o he luid. This sys em is composed by he ollowing h ee pa ial di e en ial equa ions: ⎧ ⎨ ⎩ ∂ h−di (h∇(h+z)) = ( ,x), ∂ z=Kz+sc −ehm| |n, ∂ (ch)+di (ch ) =ehm| |n−sc. (2) The i s equa ion is a mass conse a ion law o he luid and desc ibes he e olu ion o he luid heigh , deno ed by h, and he luid is anspo ed a speed := −∇(h+z). The e m is a sou ce e m ep esen ing an ex e io sou ce o wa e , like he ain. The second equa ion models he ime e olu ion o he bo om opog aphy (deno ed by z). The cons an Kis he cons an o c eep, whe eas eand sa e, espec i ely, he cons an s o he incision law and he sedimen a ion a e. As in Howa d and Ke by (1983), we will suppose ha he incision law depends on he powe o he no m o he wa e eloci y and on he powe o he wa e heigh . Thanks o his hypo hesis, as long as he wa e heigh and eloci y do no anish, he e osion a e is posi i e. Howe e , in p ac ice he e osion s a s i he wa e eloci y is high enough o b eak he cohesion o he soil. Thus, a h eshold e ec could be in oduced in he model. This e ec could be easily added o he nume ical scheme o he model, bu i would inc ease signi ican ly he di icul y o i s ma hema ical s udy. The e o e, we choose o igno e he h eshold e ec in he e osion p ocess. This app oxima ion is jus i ied in he egime ha we s udy, because he wa e heigh and wa e eloci y a e no close 20 Page 4 o 55 Fig. 1 Ske ch o he ma hema ical amewo k o s abili y analysis: The wa e is lowing on a il ed plane o ze o. The las equa ion is he conse a ion equa ion o he sedimen s dissol ed in he luid. This concen a ion, deno ed by c, is he a e age o he concen a ion o e he heigh o he luid and hus is gi en in g am pe squa e me e . The igh -hand side is he sou ce e m, which ep esen s he exchange o sedimen s be ween he g ound and he luid, caused by he e osion and he sedimen a ion. The model is se in wo dimensions, he a iables a e he ime ∈R+and he posi ion (x,y), which belongs o a domain ⊂R2. In Chen e al. (2014a), he au ho s e iewed a ious landscape e olu ion models and ocused on sys em (2). They lis ed some open ma hema ical p oblems like local exis ence in ime, egula i y o solu ions o s abili y. They pe o med a nume ical s udy on he e osion o a Gaussian shaped hill o demons a e he abili y o he model o exhibi pa e n o ma ion and s udied he in luence o some pa ame e s on he complexi y o hese pa e ns. In Leb un e al. (2018), he au ho s pe o med nume ical simula ions in a mo e p ac ical con ex whe e he ini ial opog aphy is a eal one like he ones ound in La Reunion o Madei a islands: They ook a pa icula ca e o he isualiza ion o landscape e olu ion h ough a pa icula colo iza ion p ocess. The simula ions show he capaci y o he model o desc ibe a ealis ic e olu ion in ime o he i e s and gullies on hese landscapes, p o ided i s pa ame e s a e p ope ly chosen. No e ha he de achmen limi hypo hesis is alid o bed ock i e s and mus be app op ia ely modi ied o allu ial i e s, whe e sedimen s o m a laye o loose ma e ial. In his la e case, o he classes o models may be conside ed like he shallow wa e equa ions coupled wi h Exne equa ions o he anspo o sedimen s: See Fe nández-Nie o e al. (2017) o a o mal de i a ion o hese models om bilaye - ype models (one laye o he luid and one laye o he sedimen ) and Fe nandez- Nie o e al. (2014); Escalan e e al. (2021) o a ious nume ical s a egies o pe o m simula ions o bedload sedimen anspo . The aim o his pape is o explo e he mechanism o pa e n o ma ion in he soil caused by wa e low and o p esen he o m and he equency o appa i ion o hese pa e ns. In landscapes, hese pa e ns a e channels, bed i e s, alley. We will ocus on a simple amewo k whe e he ini ial bo om su ace is a il ed plane. This su ace is co e ed by a laye o wa e , he wa e low om he op o he plane. The si ua ion is p esen ed in Fig. 1, whe e he il ed plane is seen om he side. 123 Page 5 o 55 20 Al hough i is a less ealis ic amewo k han he landscape e olu ion desc ibed in Leb un e al. (2018) whe e eal ini ial opog aphies a e conside ed, i is su icien o s udy he eme gence o pa e ns on he su ace. No e ha expe imen s o e osion on a la su aces we e conduc ed in Gué in e al. (2020) wi h a hin ilm low which e odes blocks o plas e o sal . The main ou comes o his pape a e he ollowing: •We es ablish he well-posedness o Sys em (2) when he ini ial luid heigh does no anish. •We s udy he spec al s abili y o cons an s a es when he bo om is an inclined plane and cha ac e ize igo ously he spec al ins abili y h ough a non- dimensional numbe , he so-called channeliza ion index. •We pe o m di ec nume ical simula ions o Sys em (2) o ealis ic expe imen al da a and on imescales associa ed wi h he e osion phenomenon. We ackle he p oblem o se e e CFL es ic ions due o e y sho imescales associa ed wi h he luid low. We p o ide he ea e a sho desc ip ion o ou main esul s. Fi s , we p o e he well-posedness cha ac e o Sys em (2) locally in ime unde some assump ions on he da a and he pa ame e s. We p o e he exis ence o solu ions in sui able Sobole spaces. In o de o simpli y no a ions, we will deno e o any T>0 and Ha Hilbe space L2 T(H):= L2((0,T);H);CT(H):= C((0,T);H) he space o unc ions u: → u( ,.) ∈H ha a e, espec i ely, L2in eg able o con inuous in ime on he in e al (0,T). We p o e he ollowing esul : Theo em 1.1 Le m >0,n >3o n =2,K >0, and T0>0. Le us ix wo cons an s luid heigh s h e >hmin >0. Suppose ha he ini ial da a h0,z 0,c 0sa is y h0−h e ,z0∈Hk+1(R2), c0∈Hk,h0(x)≥2hmin ∀x∈R2. wi h k =3. Suppose ha ∈L2 T0(Hk),Kh min −||h0||2 L∞≥0. Then he e exis s 0<T<T0such ha Sys em (2) admi s a unique solu ion (h,z,c)wi h h−h e ,z∈L2 T(Hk+2)∩CT(Hk+1), c∈CT(Hk). The p oo o Theo em 1.1 is based on ene gy es ima es and on a ixed poin a gu- men . The p oblem o he well-posedness o his sys em was aised in he conclusion o Chen e al. (2014a). We p o e ha i is well posed locally in ime and unde he assump ion ha he luid heigh does no anish. Wi h his assump ion, he equa ions on wa e heigh and wa e concen a ion a e pa abolic, and his p ope y is used in he p oo , o he ene gy es ima es. This alida es he model in he si ua ion whe e he luid heigh is close o a cons an , hence, a om anishing. This will be he case in ou s abili y s udy and ou nume ical simula ions. The case o anishing luid heigh is mo e in ol ed as he model becomes degene a ed pa abolic and he egula izing 20 Page 6 o 55 e ec on he luid heigh , and hus on he luid eloci y, is los : one has o conside weake classes o solu ions. Mo e p ecisely, d opping he zdependence, he equa ion on h eads ∂ h=hh+|∇h|2. In B ändle and Vázquez (2005), he au ho s p o e he well-posedness o his equa ion wi hin he class o bounded, con inuous and nonnega i e iscosi y solu ions. This is beyond he scope o his pape o couple his ype o solu ions wi h he o he equa ions o he model, and we lea e his p oblem open o u u e s udies. An o he in e es ing ques ion is he nume ical simula ion o he model in he p esence o d y a eas: This will be ca ied ou in a o hcoming wo k. The second pa o he pape is dedica ed o pa e n o ma ion when he ini ial opog aphy is an inclined plane and he bo om su ace is weakly e oded. Ou aim is o iden i y ins abili y mechanisms ha could explain he o ma ion o pa e ns. Fo ha pu pose, we linea ize Sys em (2) a ound a cons an s a e and we s udy he condi ions o spec al ins abili y and he na u e o he ins abili ies. We expec ha when he sys em is spec ally uns able o some pa ame e s and wa e ec o s, a sligh pe u ba ion o he sys em wi h hese uns able modes will g ow up and lead o he o ma ion o pa e ns in he soil. S abili y s udies ha e been done p e iously, o o he landscape e olu ion models. The pape s Smi h and B e he on (1972) and Loewenhe z (1991) analyze a model o wo equa ions, whe e he wa e is supposed o be a equilib ium. This model has s eady solu ions o which he soil heigh can be conca e in some a eas and con ex in o he a eas, depending on he sedimen discha ge law. They show ha he linea ized sys em is s able in he con ex pa s and uns able in he conca e pa s, wi h a s onge ins abili y in he ans e se di ec ion. In hei model, hey use he sedimen anspo law; hus, i is no he same amewo k as in he model (2). No e ha in a couple o ecen pape s Anand e al. (2020) and Bone i e al. (2020), he ollowing sys em o 2 PDEs was conside ed: ∂ z=Kz−e(h H)m|∇z|n+U,∂ h=di (h 0∇z |∇z|)+R,(3) whe e R,Uand V0a e cons an s. A nume ical scheme is designed in Anand e al. (2020) o (3) whe e he ime de i a i e o he luid heigh is neglec ed wi h es cases whe e he ini ial bo om opog aphy is py amidal. I is ound ha a channeliza ion index CI=em+n KnU1−n d i es he o ma ion o channels: The numbe o channels and hei b anching inc ease wi h CI. This analysis is comple ed by a spec al s abili y analysis o a spa ially non- homogeneous s eady s a e whe e he opog aphy is a hillslope which is di ided in he middle. I is ound nume ically ha he e exis s a c i ical C0 Isuch ha he s eady s a e is s able i CI≤C0 Iand uns able o he wise. 123 Page 7 o 55 20 The spec al s udy ca ied ou in his pape is new and ou analysis p o ides some explana ions o he o ma ion o pa e ns in landscapes. The appea ance o channels on he la plane is indeed he ini ial s age o de elopmen o he o ma ion o alley and i e s in landscapes. The s abili y o he sys em depends on he pa ame e s; in pa icula , he cons an o c eep Kplays an impo an ole in his s udy. We show ha he e is a c i ical alue ¯ Ksuch ha i K≥¯ K, and i ano he condi ion on pa ame e s is sa is ied, hen he sys em is spec ally s able a all equencies. I K<¯ K, hen he e exis some wa e numbe s and ec o s o which he sys em is spec ally uns able. Mo eo e , he ins abili ies g ow as he wa e ec o s o he pe u ba ions poin s in he di ec ion ans e se o he low, which explains he o ma ion o gullies and channels aligned wi h he di ec ion o he luid low. We hen eco e quali a i ely he esul s o Bone i e al. (2020). Finally, ou s abili y s udy is comple ed by di ec nume ical simula ions, which illus a e he appea ance o pa e ns o he nonlinea sys em. The space and ime scales o he model can ake a la ge ange o alues, depending o he en i onmen . On eal landscapes, he domain size can be measu ed in kilome e s, wi h a e y slow e osion a e, in he o de o 20 −200 millime e s pe housand yea . In he expe ience on sal and plas e made in Gué in e al. (2020), he domain has a size o he o de o en cen ime e s, whe eas he e osion speed is a ound one millime e pe hou (so, much as e han in eal landscape) and he luid eloci y is 1 m pe second. Pa ame e s chosen in he nume ical simula ions a e based on hese expe imen s. As in Anand e al. (2020); Bone i e al. (2020), we ha e obse ed ha dec easing K( espec i ely, inc easing he channeliza ion index CI) ein o ces he channeliza ion p ocess. No e ha we ha e ocused he e on he o ma ion o channels: Unlike simula ions made in Leb un e al. (2018) whe e he ini ial s a e is a ma u ed landscape, we s a om a simple s a e and he landscape e ol es by himsel in he simula ions. The pape is o ganized as ollows: Fi s , in Sec . 2, we desc ibe he model and he associa ed sys em o equa ions. Then, Sec .3is de o ed o he p oo o he well- posedness cha ac e o he sys em in sho ime, (see Theo em 1.1). Nex , in Sec .4we ca y ou a spec al s abili y analysis o he Sys em (2) linea ized abou a s a iona y solu ion. These s abili y esul s a e compa ed o di ec nume ical simula ions o he nonlinea sys em (2) in Sec .4.2. Finally, Sec . 6d aws a b ie conclusion o he pape and p o ides some u u e pe spec i es. 2 The Landscape E olu ion Model In his sec ion, we in oduce he landscape e olu ion model conside ed in his pape . This is a sys em o h ee pa ial di e en ial equa ions o he luid heigh h, he bo om opog aphy zand he sedimen concen a ion c. E olu ion o opog aphy. The e olu ion o he bo om opog aphy zis gi en by: ∂ z=Kz−E+S. 20 Page 8 o 55 The unc ions E=E( ,x,y)and S=S( ,x,y) ep esen he e osion speed o he soil and he sedimen a ion speed, espec i ely, wi h ≥0, (x,y)∈R2. The pa ame e K>0 is a cons an , and he e m Kzmodels he c eep o he soil. This phenomenon is a slow di usi e mo emen o he soil which occu s a la ge ime and space scales. This mo emen is caused by se e al p ocesses, such as he g a i a ional low o he soil, wind, ain splash, expansions and con ac ions o he soil due o eeze– haw, we –d y and ho –cold cycles, o biological ac i i y. In su icien ly e oded landscape, he e a e gene ally no many sha p edges, and his c eep e m, which ends o smoo h he bo om su ace, models his phenomenon. We shall see ha his e m plays a signi ican ole in he well-posedness o he model. Howe e , he c eep e ec is no supposed o be ele an in he o ma ion o pa e ns as i is a sho ime e ec and should be supposed o be small in compa ison wi h he e osion and sedimen a ion e ms. The e osion o he su ace is caused by he shea s ess and he ic ion o he wa e low. Assuming ha he luid eloci y is cons an ac oss he luid laye , his amoun s o conside ha he e osion inc eases wi h he (no m o he) wa e eloci y, and wi h he wa e discha ge Q, as in Howa d and Ke by (1983): E( ,x,y)=αQμ ν. Consequen ly, as Q=h , we will suppose ha he e osion speed depends on he powe o he no m o he wa e eloci y and on he powe o he wa e heigh . Thus, we se E( ,x,y)=eh( ,x,y) Hm| ( ,x,y)| Vn , whe e eis he e osion speed in he condi ions h=Hand | |=Vwi h H,V>0 ha , espec i ely, ep esen a e e ence luid heigh and luid eloci y. The sedimen a ion occu s when he concen a ion o sedimen s in wa e is high enough. The sedimen a ion speed inc eases wi h he concen a ion o sedimen in he luid. Fo he sake o simplici y, we suppose ha his speed is p opo ional o he concen a ion and we se : S( ,x,y)=sc( ,x,y) csa , wi h s he speed o sedimen a ion in he e e ence condi ion c=csa . The e o e, he soil ele a ion e ol es acco ding o he equa ion: ∂ z=Kz−eh( ,x,y) Hm| ( ,x,y)| Vn +sc( ,x,y) csa .(4) Page 9 o 55 20 The landscape e olu ion model. We comple e Equa ion (4) wi h wo e olu ion equa ions o he luid heigh hand sedimen concen a ion c. The mass conse a ion law o he luid eads ∂ h+di (h ) = ,(5) whe e is a sou ce e m, modeling an incoming low in a channel o he ain o e he bo om. On he o he hand, he mass conse a ion law o he sedimen eads ∂ (hc)+di (ch ) =ρs(E−S), (6) whe e ρsis he olume ic mass densi y o he sedimen s and is cons an . In o de o close Sys em (4), (5), (6), we need o w i e an equa ion o he luid eloci y. One possibili y would be o w i e a shallow wa e - ype model wi h an e olu ion equa ion o he momen um h . We a he choose he simple closu e =−μ∇(h+z), (7) whe e μ>0 is some cha ac e is ic luid eloci y and ∇(h+z)is he g adien o he luid su ace ele a ion. Sys em (4), (5), (6), (7) is closed, and we shall conside i s well-posedness in Sec .3. We a e also in e es ed in he pa e n o ma ion a he su ace o he soil. Fo ha pu pose, we ha e chosen o explo e he case o wa e lowing down an inclined plane. This si ua ion was conside ed expe imen ally in Gué in e al. (2020). The domain ⊂R2has leng h Lxand wid h Ly:=[0,Lx]×[0,Ly]. Deno e θ he inclina ion o he plane. We can decompose he bo om opog aphy zas z( ,x,y)=(Lx− x) an θ+˜z( ,x,y)whe e ˜zis he e oded heigh o he soil. Thus, he luid eloci y is w i en as ( ,x,y)=μ( an θ,0)−μ∇(˜z+h). Consequen ly, omi ing he ilde o e z, Sys em (4), (5), (6), (7) admi s he new o m: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ h+μ an θ∂xh=μdi (h∇(h+z)), (8a) h∂ c+μh an θ∂xc=μh∇(h+z).∇c+ρseh( ,x) Hm| ( ,x)| Vn −ρssc( ,x) csa ,(8b) ∂ z=Kz−eh( ,x) Hm| ( ,x)| Vn +sc( ,x) csa .(8c) 3 Well-Posedness o he Landscape E olu ion Model In his sec ion, we s udy he exis ence and uniqueness o solu ions o he sys- em (4), (5), (6), (7), locally in ime. 20 Page 16 o 55 The classical heo y o pa abolic equa ions and o linea anspo equa ions p o- ides condi ions o ob ain a well-posed sys em o equa ions. Fo he wo pa abolic equa ions (25a) and (25b), we s a e he ollowing esul (see Che ie and Milani 2012 o mo e de ails): P oposi ion 3.7 Assume ha h0−h e ,z0∈Hk+1(R2), ,sc i−ehm i| i|n∈L2 T(Hk), hi∈L∞([0,T]×R2)wi h hi≥hmin and ∇hi,∇zi∈L1 T(Hk). Then he e exis s a unique solu ion (hi+1,zi+1) o he equa ions (25a), (25band hi+1−h e ,zi+1∈ L2 T(Hk+2)∩CT(Hk+1(R2)). Fo he anspo equa ion (25c), we use heo em 7.2.2inMé i ie (2008) o s a e he esul : P oposi ion 3.8 Assume ha c0∈Hk(R2),∇ i∈L∞ T(Hk−1),ehm−1 i| i|n−sci/hi− ci/hi∈L1 T(Hk)and i∈L1 T(W1,∞). Then he e exis s a unique solu ion ci∈ CT(Hk) o he equa ion (25c). We p o e by induc ion ha o all i∈N, Sys em (25a) is well posed. Fi s , o i=0, he unc ions o (hi,zi,zi)a e ime independen and h0≥hmin. Thus, as T0=Tis ini e, we ha e sc0−ehm 0| 0|n∈L2 T0(Hk),h0∈L∞ T0(R2)and ∇h0,∇z0∈L1 T0(Hk). Then he e exis s a unique solu ion (h1,z1)∈L2 T1(Hk+2)∩CT1(Hk+1)o (25a) and (25b) wi h 0 <T1≤T0such ha h1≥hmin. Simila ly, he e exis s c1∈CT1(Hk) solu ion o (25c). Now, i we assume ha hi−h e ,zi∈L2 T(Hk+2)∩CTi(Hk+1)and ci∈CTi(Hk) hen, one has hi∈h e +CTi(H2)⊂L∞([0,T]×R2). Mo eo e , we ha e ∇hi,∇zi∈CTi(Hk)⊂L1 Ti(Hk)and ci∈CTi(Hk)⊂L2 Ti(Hk). Finally, one has hm i| i|nHk≤Chm−3 iL∞hiL∞+∇hiHk+∇ziHkn+3∈L∞ Ti(Hk)⊂L2 Ti(Hk). Thus, he assump ions o P oposi ion 3.7 a e sa is ied and he e exis s a unique solu ion hi+1,zi+1o (25a,25b) such ha hi+1−h e ,zi+1∈L2 Ti+1(Hk+2)∩ CTi+1(Hk+1(R2)) wi h 0 <Ti+1≤Tisuch ha hi+1≥hmin. The exis ence o a solu ion ci+1∈CTi+1(Hk) ollows simila ly. P oposi ion 3.9 (Uni o m bounds) Deno e Ei( )=hi−h e L2+∇hiHk+ziHk+1+ciHk. The e exis s B >0and 0<T∗≤T independen o i ∈Nsuch ha , o all i ∈N sup ∈[0,T∗] Ei( )≤B,and hi( ,x)≥hmin,∀x∈R2. Mo eo e , T∗ 0∇hi2 Hk+1+∇zi2 Hk+1≤C(T∗)B,∀i∈N∗. Page 17 o 55 20 P oo We p oceed by induc ion. One has easily E0( )=E(0)≤Band h0≥hmin since h0,z0,c0a e ime independen . Following he s a egy used o de i e a p io i es ima es, one can p o e ha o some ε>0, o all ∈[0,T], d d E1( )+ε∇h12 Hk+1+∇z12 Hk+1≤C(ε)( 2 Hk+E0( )α)E1( ), (26) o some cons an C(ε) depending only on εand h0. By in eg a ing Equa ion (26) wi h espec o ime, one inds: E1( )+ε 0∇h12 Hk+1+∇z12 Hk+1 ≤E(0)+C(ε)  0 2 Hk+ 0 C(ε)E0(s)αE1(s)ds. Deno e B=2E(0)+C(ε) T 0 2 Hk. We choose ˜ T∗such ha e˜ T∗C(ε)Bα≤2. Then, by applying G onwall’s lemma, one ob ains: E1( )≤B,∀ ∈[0,˜ T∗], ˜ T∗ 0∇h12 Hk+1+∇z12 Hk+1≤C(ε)1+ln(2) εB. Nex , we ha e h1=h0+ 0 di (h0∇(h1)+h1∇z0)) + . Thus, h1−h0L∞≤C(˜ B +√B ) o some cons an Cindependen o he p oblem and ela ed o Sobole injec ions, whe eas ˜ B=B+h e √B. Then, he e exis s 0<T∗≤˜ T∗such ha C(˜ B +√B )≤hmin and we deduce ha h1≥h0−hmin ≥hmin ∀ ∈[0,T∗]. This p o es he ini ial s ep o i=1. Now assume ha Ei( )≤Band hi≥hmin o all ∈[0,T∗]. The es ima es on hi+1,zi+1and ci+1a e a di ec consequence o he 20 Page 18 o 55 ene gy es ima e: d d Ei+1( )+ε∇hi+12 Hk+1+∇zi+12 Hk+1≤C(ε)( 2 Hk+Ei( )α)Ei+1( ) (27) which is p o ed by ollowing he s a egy used o de i e he a p io i es ima es. This comple es he p oo o he p oposi ion.  P oposi ion 3.10 The sequences (∂ hi)e (∂ zi)a e uni o mly bounded in L2 T∗(Hk). The sequence (∂ ci)is uni o mly bounded on L2 T∗(Hk−1). P oo Fo p∈{0,...,k}, we di e en ia e p imes he equa ions (25a), (25b)(25b) and mul iply i by ∇p∂ hi,∇p∂ ziand ∇p∂ ci, espec i ely, and in eg a e in space. Using P oposi ion 3.9, we ob ain bounds o ∂ hiL2 T∗(Hk),∂ ziL2 T∗(Hk)and ∂ ciL2 T∗(Hk−1).  3.5 Con e gence o he Sequences In wha ollows, we deno e T∗=Tin o de o simpli y he no a ions. P oposi ion 3.11 The sequences (hi)i∈N,(∂ hi)i∈Nand (zi)i∈N,(∂ zi)i∈Na e Cauchy sequences in L2 T(L2). The sequences (ci)i∈Nand (∂ ci)i∈Na e Cauchy sequences in CT(L2). P oo In he ollowing, we deno e he quan i ies o he o m i+1− iby δ i. Fo all i≥1, he equa ions o δhi,δzi,δcia e w i en as: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ δhi−di (hi∇δhi+δhi−1∇hi)−di (hi+1∇δzi+δhi∇zi−1)=0,(28a) ∂ δzi−Kδzi=sδci−ehm i(| i|n−| i−1|n)+| i−1|n(hm i−hm i−1),(28b) ∂ δci+δ i−1∇ci+1+ i−1δ∇ci =ehm−1 i(| i|n−| i−1|n)+| i−1|n(hm−1 i−hm−1 i−1) −(s+ )ci hi−ci−1 hi−1. (28c) Bounds on δhi,δziand δci: We mul iply he equa ion (28a)byδhiand in eg a e o e space: 1dδhi2 L2+R2(hi∇hi+1−hi−1∇hi)∇δhi+R2(∇zihi+1−∇zi−1hi)∇δhi=0. Page 19 o 55 20 Fi s , on he one hand, we ha e R2 (hi∇hi+1−hi−1∇hi)∇δhi=R2 hi(∇δhi)2+δhi−1∇hi∇δhi ≥hmin∇δhi2 L2+R2 δhi−1∇hi∇δhi. On he o he hand, we ha e he es ima e: R2 (∇zihi+1−∇zi−1hi)∇δhi=1 2R2∇zi∇(δhi)2+R2 δzi−1hi∇δhi =−1 2R2 zi(δhi)2+R2 δzi−1hi∇δhi ≤Cδhi2 L2+hiL∞δzi−1L2∇δhiL2 ≤Cδhi2 L2+1 2hmin δzi−12 L2+hmin 2∇δhi2 L2. Consequen ly, we ob ain: d d δhi2 L2+hmin∇δhi2 L2≤Cδhi2 L2+Cδhi−12 L2+Cδzi−12 L2.(29) We p oceed simila ly o δzi. By using Equa ion (28b), one inds: d d δzi2 L2+2K∇δzi2≤s(δci2 L2+δzi2 L2)+2eR2(hm i| i|n−hm i−1| i−1|n)δzi. In o de o bound he igh -hand e m, we use he inequali y ||x|n−|y|n|≤n|x− y|max(|x|,|y|)n−1: hm i| i|n−hm i−1| i−1|n=hm i(| i|n−| i−1|n)+| i−1|n(hm i−hm i−1) ≤nhm i| i− i−1|max( i−1L∞, iL∞)n−1 +m hmin | n i−1|hi−hi−1|max(hiL∞,hi−1L∞)m. Consequen ly, as hi,hi−1and i, i−1a e uni o mly bounded in L∞([0,T]×R2): R2 (hm i| i|n−hm i−1| i−1|n)δzi≤C( i− i−1L2+hi−hi−1L2)δziL2 ≤C(∇δhi−1L2+∇δzi−1L2+δhi−1L2)δziL2 ≤C(∇δhi−12 L2+∇δzi−12 L2)+δhi−12 L2+δzi2 L2). Thus, 20 Page 20 o 55 d d δzi2 L2+2K∇δzi2 L2≤Cδzi2 L2+C(∇δhi−12 L2+∇δzi−12 L2+δhi−12 L2). (30) Finally, o δci, we use he same me hod as be o e wi h he equa ion (28c) and we ob ain: 1 2 d d δci2 L2=−(δ i−1∇ci+1+ i∇δci)δci +e(hm i| i|n−hm i−1| i−1|n)δci−(s+ )ci hi−ci−1 hi−1δci =−δ i−1∇ci+1δci−1 2di ( i)(δci)2 +e(hm i| i|n−hm i−1| i−1|n)δci−(s+ )ci hi−ci−1 hi−1δci ≤δ i−1L2∇ci+1L∞δciL2+∇ iL∞δci2 L2 +C(∇δhi−12 L2+∇δzi−12 L2+δhi−12 L2) +s+ H2 hmin ciL2+ci−1L2δciL2.(31) Then we add he inequali ies (29), (30) and (31) and in eg a e on [0, ]wi h 0 ≤ ≤T: δhi2 L2+δzi2 L2+δci2 L2+hmin  0∇δhi2 L2d +2K 0∇δzi2 L2d ≤δh02 L2+δz02 L2+δc02 L2+C 0δhi2 L2+δzi2 L2+δci2 L2d +C 0δhi−12 L2+δzi−12 L2+δci−12 L2d +C 0 (∇δhi−12 L2+∇δzi−12 L2)d . Page 21 o 55 20 We apply he G onwall lemma and ob ain, o all ∈[0,T], δhi2 L2+δzi2 L2+δci2 L2+hmin  0∇δhi2 L2d +2K 0∇δzi2 L2d ≤Cδh02 L2+δz02 L2+δc02 L2 + 0δhi−12 L2+δzi−12 L2+δci−12 L2d + 0 (∇δhi−12 L2+∇δzi−12 L2)d eC . Wi h his inequali y, we deduce by induc ion on i ha ∀i∈N, δhi2 L2 T(L2)+δzi2 L2 T(L2)+δci2 L∞ T(L2) ≤CTe CTi i!δh02 H1+δz02 H1+δc02 L2. Consequen ly, he se ies δhi2 L2 T(L2),δzi2 L2 T(L2)and δci2 L∞ T(L2)con- e ge; hus, he sequences (hi)i∈N,(zi)i∈N,(ci)i∈Na e Cauchy sequences in he equi ed spaces. Bounds on ∂ δhi,∂ δziand ∂ δci: Like he es ima es in P oposi ion 3.10,weuse he sys em (28a) and P oposi ion 3.9 o ob ain he bounds.  By P oposi ion 3.11, he e exis s h−h e ,z∈L2 T(L2)such ha hi−h e con e ges o h−h e and zicon e ges o zin L2 T(L2).As(hi−h e )and (zi)a e uni o mly bounded in L2 T(Hk+2), we ob ain by in e pola ion ha ∀1/2<θ<1, ∀i>j∈N, T 0hi−hj2 Hθ(k+2)d ≤T 0hi−hj1−θ L2hi−hjθ Hk+2d ≤hi−hj1−θ CT(L2)T 0hi−hjθ Hk+2d ≤hi−hj1−θ CT(L2)T2/(2−θ)hi−hjθ L2 T(Hk+2) ≤hi−hj1−θ CT(L2)T2/(2−θ)(2C1)θ−→ n→+∞ 0. The e o e, ∀k/2<s<k, he sequences (hi−h e )and (zi)a e Cauchy sequences in L2 T(Hs+2); hus, h−h e ,z∈L2 T(Hs+2). Mo eo e , ∂ hicon e ges o ∂ hin L2 T(L2) and (∂ hi)is uni o mly bounded in L2 T(Hk), simila ly o ∂ z. Consequen ly, ∀s<k, ∂ h,∂ z∈L2 T(Hs); hus, by P oposi ion 3.5,h−h e ,z∈CT(Hs+1). Finally he a p io i es ima es on hand zallow o conclude ha h−h e ,z∈L2 T(Hk+2)∩CT(Hk+1). In pa icula , as k+1=4, (hi−h e ),(∇hi),(∇2hi),(∂ hi)and (zi),(∇zi),(∇2hi), (∂ zi)con e ges in C([0,T]×R2). 20 Page 22 o 55 Now we conside (ci)i∈N. By P oposi ion 3.11, he e exis s c∈CT(L2)limi o (ci)i∈Nin his space. We know ha (ci)i∈Nis uni o mly bounded in CT(Hk),soby in e pola ion: ∀0<θ<1, ∀i>j∈N, sup ∈[0,T]ci−cjHθ(k)≤sup ∈[0,T]ci−cj1−θ L2sup ∈[0,T]ci−cjθ Hk ≤ci−cj1−θ CT(L2)(ciCT(Hk)+cjCT(Hk))θ ≤Cci−cj1−θ CT(L2)−→ n,m→+∞ 0. The e o e, (ci)i∈Nis a Cauchy sequence and hus con e ges o cin he space CT(Hs), o all s<k. Mo eo e , (∂ ci)con e ges o ∂ cin CT(L2)and is uni o mly bounded in CT(Hk−1), so i con e ges o cin CT(Hs−1). And we conclude by he a p io i es ima e on c ha c∈CT(Hk). To conclude, (ci)i∈N,(∂ ci)i∈Nand (∇ci)i∈Ncon e ge in CT(R2); hus, we can ake he limi in he equa ions, and (h,z,c)is solu ions o Sys em (9a). This concludes he p oo o he exis ence. 3.6 Uniqueness P oposi ion 3.12 Le (h1,z1,c1)and (h2,z2,c2)be wo solu ions o Sys em (9a)sa - is ying he hypo heses o Theo em 1.1. Then ∀ ∈[0,T]: h1( )−h2( )2 L2+z1( )−z2( )2 L2+c1( )−c2( )2 L2 ≤h0 1−h0 22 L2+z0 1−z0 22 L2+c0 1−c0 22 L2eCT In pa icula , when he ini ial condi ions a e he same o bo h solu ions, hese solu ions a e he same. Consequen ly, his p oposi ion shows he uniqueness o he solu ion (h,z,c)o he heo em. P oo We i s w i e he equa ions e i ied by δh:= h1−h2,δz:= z1−z1and δc:= c1−c2: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ δh−di (h1∇δh+δh∇h2)−di (h1∇δz+δh∇z2)=0 (32a) ∂ δz−Kδz=sδc−ehm 1(| 1|n−| 2|n)+| 2|n(hm 1−hm 2)(32b) ∂ δc+δ ∇c2+ 2δ∇c=ehm−1 1(| 1|n−| 2|n)+| 2|n(hm−1 1−hm−1 2) −(s+ )c1 h1−c2 h2(32c) Then he bound is ob ained wi h a simila p ocess as in he p oo o P oposi ion 3.11. 123 Page 23 o 55 20 4 Spec al S abili y o Cons an S a es In his sec ion, we conside he low o e a opog aphy ha is an inclined plane a ime =0. We assume ha =0 and we s udy he spec al s abili y o cons an s a es. We expec ha ins abili y will p o ide a mechanism o pa e n o ma ion. We i s w i e Sys em (8a) in a non-dimensional o m and hen linea ize his sys em a ound cons an s a es. Then we explo e nume ically he s abili y o he sys em. Finally, we ca y ou he spec al s abili y analysis by using Rou h–Hu wi z heo em: This p o ides necessa y and su icien condi ions o cons an s a es o be spec ally s able. Howe e , hese condi ions do no p o ide any insigh on he na u e o he ins abili ies. We comple e his analysis by an asymp o ic expansion o he spec um a ound he o igin and in he high- equency egime. 4.1 Non-dimensionaliza ion and Linea iza ion o he Sys em We w i e Sys em (8a) in a non-dimensional o m in o de o iden i y he impo an pa ame e s. We in oduce se e al cha ac e is ic quan i ies: Zis a cha ac e is ic e oded heigh , Ha cha ac e is ic wa e heigh , La cha ac e is ic wa eleng h and Ta cha ac- e is ic ime. We chose T o be he necessa y ime o e ode he soil o a heigh Z, wi h an e osion speed e. Thus, T e i ies eT =Z.As =μ an θe1−μ∇(z+h),we ix he cha ac e is ic wa e eloci y V=μ. We in oduce he dimensionless a iables: h:= h H,z=z Z, := V,c:= c csa x:= x L, := e Z . In o de o simpli y he no a ions, we will assume H=Z. Then, d opping he p imes, Sys em (8a) is w i en as: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ h+ZV an θ eL ∂xh=Z2V eL2di (h∇(h+z)), h∂ c+ZV an θ eL h∂xc=ZV eL2h∇(h+z).∇c+ρs csa hm| an θe1−Z L∇(h+z)|n−s e ρs csa c, ∂ z=ZK eL2z−hm| an θe1−Z L∇(h+z)|n+s ec. To simpli y he equa ions, we se Z L=e V, and we de ine α:= Z L=e V,K:= ZK eL2= K LV . The sys em eads: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ∂ h+ an θ∂xh=αdi (h∇(h+z)), h∂ c+ an θh∂xc=αh∇(h+z).∇c+ρs csa hm| an θe1−α∇(h+z)|n−ρss csa ec, ∂ z=Kz−hm| an θe1−α∇(h+z))|n+s ec. (33) 20 Page 24 o 55 The s a iona y s a es o Equa ion (33) o a la su ace, deno ed by (h,c,z), e i y: ∀ ∈R+,∀(x,y)∈, ⎧ ⎪ ⎨ ⎪ ⎩ h( ,x,y)=h>0, c( ,x,y)=c=e shm annθ>0, z( ,x,y)=0. This means ha he e osion and he deposi ion p ocess equilib a e each o he and he bo om is no e oded, whe eas he luid heigh is a cons an . Le (h+h,c+c,z), wi h |h|,|c|,|z|1, be a small pe u ba ion o he con- s an s a e, and solu ion o Sys em (33). Then, a i s o de , his solu ion e i ies he ollowing linea sys em: ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ h+ an θ∂xh=αh(h+z) ∂ c+ an θ∂xc=ρs csa s emc h2h−αnc h an θ∂x(h+z)−c h ∂ z=Kz−s emc hh−αnc an θ∂x(h+z)−c(34) We deno e =(h,c,z)T. Then e i ies he equa ion ∂ =A0 +A1∂x +A2 wi h: A0=⎡ ⎢ ⎢ ⎢ ⎣ 000 amc h−a0 −amc ¯ρs ah ¯ρs 0 ⎤ ⎥ ⎥ ⎥ ⎦,A1=⎡ ⎢ ⎢ ⎢ ⎣ − an θ00 −αanc an θ− an θ−αanc an θ αanh c ¯ρs an θ0αanh c ¯ρs an θ ⎤ ⎥ ⎥ ⎥ ⎦,A2=⎡ ⎣ αh0αh 000 00K⎤ ⎦, whe e we ha e deno ed a=sρs ehcsa and ¯ρs=ρs csa . We apply he Fou ie ans o m in space and he equa ion e i ied by ˆ , he Fou ie ans o m in space o ,is: ∂ ˆ =A0+iξA1−(ξ2+η2)A2ˆ := A(ξ, η) ˆ . Consequen ly, in o de o s udy he s abili y o he sys em, we ha e o de e mine he sign o he eal pa o he eigen alues o he ma ix A(ξ, η). These eigen alues a e deno ed by λi(ξ, η) wi h i∈{1,2,3}and he associa ed eigen ec o s a e deno ed by Vi(ξ, η). The exp essions o λia e no explici : In he nex subsec ion, we compu e nume ically he s abili y o he sys em. Then Sec . 4.3 gi es a s abili y esul on he domain. This esul is comple ed wi h he asymp o ic s udy o he eigen alues a low (|ξ|2+|η|21) and high (|ξ|2+|η|21) equencies. 123 Page 25 o 55 20 Fig. 2 S abili y diag ams in he plane (ξ, η) ⊂R2, o K>0. The sys em is uns able in he ed a ea and s able in he g een a ea (Colo igu e online) 4.2 Nume ical Explo a ion o S abili y In his sec ion, we explo e nume ically he s abili y o he sys em (34). Fo ha pu pose, we ha e compu ed nume ically he h ee eigen alues o he ma ix A(ξ, η), and he sys em is s able i and only i he eal pa s o hese eigen alues a e nega i e. Since A(ξ, −η) =A(ξ, η) and A(−ξ,η) =A(ξ, η), he eal pa o he spec um emains unchanged unde he ans o ma ion ξ→−ξand η→−η. Thus, we examine he beha io o he sys em in he op igh qua e o he plane. The choice o pa ame e s o hese compu a ions is he same as in Sec .5(unless o he wise speci ied), see Table 1 o hei alues. In Fig.2, we ha e p esen ed an illus a ion o he s abili y o he sys em when K>0, whe e Ke=5×10−4 3600 m2s−1. The domain ep esen ed is a bounded subse o he plane (ξ, η) ⊂R2, and he colo ep esen s he s abili y: The sys em is s able in he g een a ea and uns able in he ed a ea. We clea ly see he s abilizing e ec o he c eep e ec : When Kis highe , he s able a ea is la ge . I seems ha he uns able a ea is bounded. This will be con i med by P oposi ion 4.2. Then, in Fig.3 he e is no c eep e ec : K=0, and we ep esen a ious s abili y diag ams o se e al alues o he a io n/m.The alueo mis he same as in Table 1: m=1.6, and n a ies be ween m/2 and 10m. We obse e ha he beha io o he sys em changes wi h he a io m/n, bu he uns able a ea always seems o be unbounded. When nmas in Fig.3a, he s able a ea is bigge han he uns able a ea, and he sys em is uns able only o pe u ba ions o ans e se dominan di ec ion. When n≈m,asinFig.3c, he sys em is s able only o longi udinal pe u ba ions. Finally, when n<m he sys em seems o be uns able a all equencies, as in Fig.3d. 20 Page 32 o 55 Table 1 Pa ame e s o nume ical simula ions Leng h o he domain Lx40 cm Wid h o he domain Ly10 cm Cha ac e is ic wa e speed V=μ an θ1m/s Ini ial wa e heigh h00.5mm Ini ial sedimen concen a ion c0317 g/m3 Exponen o ic ion o e hm1,6 Exponen o ic ion o e n3,2 Densi y o he sedimen s ρs2.17 ×106g/m3 Concen a ion o sa u a ion csa 3.17 ×105g/m3 E osion speed e0.5 mm/hou Sedimen a ion speed se/2000 Angle o he plane θ39◦ 5 Di ec Nume ical Simula ions In his sec ion, we p esen some nume ical expe imen s o he e osion o a il ed plane. The pa ame e s o hese expe imen s come om physical da a. The quan i ies Lx,Ly, V,h,ρs,csa ,eand θa e chosen acco ding o he labo a o y expe imen (Gué in e al. 2020), which e odes a block o sal . The choice o alues o he exponen s mand nhas been in es iga ed many imes in he li e a u e. The alues a e chosen be ween 0 and 3, wi h an addi ional ela ion n=2m, as explained in Chen e al. (2014b) (he e he cons an nco esponds o he cons an m+nin he li e a u e). We choose n=2m, wi h nsu icien ly la ge in o de o obse e he o ma ion o channels in he simula ions. Indeed, we ound ha he e ec o digging in dep essions is ein o ced when hese exponen s a e la ge . The choice o pa ame e s is gi en in Table 1. Once he a io be ween he e osion speed and he wa e speed has been ixed, we assume ha he ime a ia ions o he wa e heigh and concen a ion a e small; consequen ly, we neglec hese a ia ions in he simula ion. Indeed, in he simula ion, he e oded heigh o he soil is o he o de o a millime e ; hus, he cha ac e is ic ime T=Z/eis o he o de o an hou . The wa e c osses he domain in 0.4 seconds; hus, he e a e ou o de s o magni ude be ween he cha ac e is ic ime o he wa e low and ha o e osion. A di ec nume ical simula ion o he ull sys em, wi h ime de i a i es, would impose a se e e CFL es ic ion: Indeed, he luid eloci y is abou 1m.s−1, whe eas he e osion a e is a ound 1mm.h−1. Since we a e in e es ed in he e osion p ocess, he na u al imescale is one hou and a ypical ime s ep would be a minu e. Howe e , he nume ical ime s ep δ is d i en by a CFL: δ ≤δx/ wa e ≈0.001si one conside a ypical mesh size δx=1mm ( o a channel o leng h 400mm). This inc eases he nume ical cos o he scheme. Ins ead, we sol e s a iona y p oblems a each ime s eps, whe e he ime s ep is de e mined by he e osion imescale. A compa ison be ween he wo esolu ion me hods is made in Appendix D, and i shows ha aking he s a iona y equa ion does no a ec he esul s. 123 Page 33 o 55 20 Fig. 7 Pe u ba ion o he ini ial su ace, on he domain [0,Lx]×[0,Ly].The ep esen ed quan i y is he di e ence be ween he la su ace and he pe u bed soil, in millime e s. This quan i y is ze o a yellow poin s and inc eases when colo is da ke (Colo igu e online) The s a iona y equa ions o he wa e heigh and concen a ion o sedimen s in wa e a e disc e ized wi h a ini e olume me hod. The scheme is gi en in Appendix D. Equa ion (8b)oncis a linea equa ion, and we disc e ize i wi h an explici Eule scheme by conside ing i as an e olu ion equa ion wi h espec o he a iable x,as he speed in his di ec ion does no anish: ∂xc+ y x ∂yc=ρs h(E−S). (37) Equa ion (8a)onhis nonlinea ; hus, i is ha de o disc e ize i . In o de o a oid an implici disc e iza ion o his equa ion, we made he choice o linea ize he equa ion. Deno ing by hn he solu ion o he equa ion a ime n, we app oxima e: di (h∇h)( n)+di (h∇z)( n)≈di (hn−1∇hn)+di (hn∇zn). As he solu ion a he p e ious ime s ep is known, he igh -hand e m is linea in hn. Thus, we can disc e ize i wi h a ini e olume scheme in wo dimensions. The jus i ica ion o he quasi-s a iona y model and he nume ical scheme a e gi en in Appendix D. We ha e chosen pe iodic bounda y condi ions in he ans e se di ec ion; he e o e, in he s abili y analysis he e a e only coun able equencies in his di ec ion. The equencies a e he ηn=2πn/Lywhe e Lyis he wid h o he domain. The bounda y condi ions a he op o he domain a e Di ichle condi ion o h,cand z. A he bo om o he domain, z is p esc ibed by a Di ichle condi ion and we suppose ha wa e lows eely. Thus, we ix Neumann condi ion o hand c. In he simula ions, he ini ial su ace is a la il ed plane wi h a small andom pe u ba ion. This su ace is p esen ed in Fig.7, and i shows a la iew o he wo- dimensional plane. The colo scale shows he heigh di e ence be ween he ac ual soil and he la plane. Thus, he yellow a eas a e he less dug pa s. 20 Page 34 o 55 Jou nal o Nonlinea Science (2024) 34 :20 Fig. 8 E oded heigh o he soil in mm. The da kes a eas a e he mos e oded a eas Fig. 9 E oded heigh o he soil in mm wi h espec o he la su ace. The da kes a eas a e he mos e oded a eas. The wid h o he channels is la ge o la ge alues o K 5.1 Simula ions Wi hou Sou ce Te m o Wa e In his pa , we p esen some esul s o simula ions when =0, as in he spec al s abili y analysis. These simula ions a e compa ed o he heo e ical esul s o s abili y. Fi s , in Fig.8we p esen s esul s o he simula ion o he sys em, when K=Ke=5×10−4 3600 m2s−1. In his case, he sys em is spec ally s able a all equencies {(ξ, ηn);ξ∈R,n∈ N}. The pic u es ep esen he e oded heigh z(T)−z0a ime T=0.25 hou and T=2 hou s. We obse e ha su ace pe u ba ions a e quickly smoo hed, and end o disappea . A e 15 minu es, we see in Fig. 8a ha he ampli ude o pe u ba ion has no dec eased ye , bu hese pe u ba ions a e smoo he han ini ially. A e 2 hou s, we see in Fig.8b ha he ampli ude o he pe u ba ions has dec eased. In Figs.9and 10, we p esen , espec i ely, he e oded su ace and he luid heigh when he sys em is uns able a some equencies. In Fig.9, we can obse e he o ma- ion o channels in he soil, in he low di ec ion. The wid h o hese channels is la ge when Kis highe , and hey ake mo e ime o appea . This can be explained by he 123 Page 35 o 55 20 Fig. 10 Wa e heigh on he domain, in millime e s. The wa e dep h is g ea e in da ke a eas Fig. 11 Disc e e equency analysis o he esul o he simula ion, when K=Ke/20 (Fig. 9a) and K=Ke/50 (Fig.9b). The quan i y ep esen ed is he no m o he disc e e Fou ie ans o m o he signal in he ans e se di ec ion s abili y analysis, as discussed below. Figu e10 has some simila i ies wi h Fig.9since he wa e ends o ill he e oded channels; he wa e dep h is la ge in he channels and smalle be ween hem. We quan i y he numbe o channels gene a ed by he simula ions in o de o alida e he s abili y analysis. Fo ha pu pose, we compu e he disc e e Fou ie ans o m o he e oded su ace a he end o he simula ion. As he main di ec ion o pe u ba ions is ans e se o he slope, we ha e calcula ed his Fou ie ans o m in his di ec ion, a ixed x.InFig.11, we p esen he no m o he disc e e Fou ie ans o m (DFT) o he esul o he simula ions 9a, 9b in he ans e se di ec ion, calcula ed a he bo om o he plane ( o x=Lx). The equencies a e he 2πN/Ly, o N∈{0,...,Ny/2}. When K=Ke/20, we can see in Fig. 11 ha he dominan equency is eached when N=2 a he bo om o he il ed plane. When K=Ke/50, we ha e con ibu ion be ween N=2 and N=5 equency Then, in o de o compa e his obse a ion wi h he s abili y analysis, we compu e nume ically he equencies which c ea e he mos uns able modes. In Fig.12,we 20 Page 36 o 55 Fig. 12 Maximum o he eal pa o he h ee eigen alues, no malized by he highe one, o (ξ, η) ∈ [0,20]×[0,300]. The mos uns able a eas a e in yellow, and in he black pa s, he h ee eigen alues ha e a nega i e eal pa ; hus, he sys em is s able (Colo igu e online) p esen nume ical compu a ions o he eigen alues o he linea ized sys em, depending on he equencies. The la ges eal pa o he h ee eigen alues is ep esen ed, o each equencies ξand ηin a bounded domain. This quan i y con ol he s abili y, he sys em is s able i and only i i is nega i e. Due o he pe iodic bounda y condi ions, he equencies allowed in he ans e se di ec ion a e he ηn=2πn/Ly, whe e Lyis he wid h o he domain. In Fig.12, he whi e lines shows hese equencies. Fo bo h cases (Fig.12a, b), he sys em is uns able and he maximum o ins abili y is eached a ξ=0, η1=2π/Ly. Thus, he sys em des abilizes in he ans e se di ec ion, and he mos des abilizing equency has pe iod one. This is o he same o de o magni ude as compu ed by he DFT o he simula ions, whe e his equency is (ξ, η) =(0,2×2π/Ly). The di e ence be ween he pe iod o 1 p edic ed by he s abili y analysis and he pe iods o 2−5 ob ained in he simula ions could come om he nonlinea e ec s o he model ha a e no aken in accoun in he s abili y analysis. 5.2 Simula ions Including a Sou ce Te m o Wa e In his pa , we p esen some esul s o simula ions when he sou ce e m is a posi i e unc ion, o obse e he e ec o ain in he model. In he ollowing simula ions, K=Ke/20. Fi s , we conside a cons an in ime and uni o m in space sou ce e m 1= 0.005mm/s. Figu e13 shows he su ace heigh (le igu e) and wa e heigh ( igh igu e) compu ed by he simula ion, a ime 0.25 hou s. We can obse e ha he wa e heigh is almos ou imes highe a he bo om o he domain (x=Lx), han a he op (x=0). As he e osion a e is p opo ional o he powe o he wa e heigh h, he su ace e odes as e a he bo om o he domain. Then, Fig.14 shows he esul s o he same simula ion, a ime 2 hou s. The ini- ial pe u ba ions ha e almos disappea ed, and some ans e se pe u ba ions ha e de eloped a he bo om o he domain. 123 Page 37 o 55 20 Fig. 13 Resul s o he simula ion when = 1, a e 2000 i e a ions Fig. 14 Resul s o he simula ion when = 1, a e 16000 i e a ions Fig. 15 Resul s o he simula ion when = 2, a e 14000 i e a ions Nex , we choose a bigge sou ce e m: 2=0.01m/s, and esul s o his simula ion a e shown in Fig. 15. We can see ha he e a e bigge pe u ba ions a he bo om o he domain han wi h he sou ce e m 1. These pe u ba ions a e also no o ally ans e se: hey undula e a li le bi in he longi udinal di ec ion, which is a di e en beha io om he case wi hou sou ce e m (Fig. 9b). Now we ake a cons an non-uni o m sou ce e m: 3(x,y)=0.005 exp −60 ∗(y−Ly/2)2 Ly2. 20 Page 38 o 55 Fig. 16 Resul s o he simula ion when = 3, a e 6000 i e a ions ha co espond o a posi i e longi udinal band o ain in he middle o he domain which dec ease exponen ially as when app oaching y=0o y=0.1. A ime 0.75 hou s, he su ace is highly e oded a he bo om o he band (Fig. 16). He e, he e olu ion o he sys em is d i en by he sou ce e m, and he e osion landscape depends mainly on his e m. These simula ion show ha he sou ce e m may quan i a i ely in luence he esul s, al hough quali a i ely hey look simila o he case wi h no sou ce e m. 6 Conclusion and Pe spec i es In his pape , we ha e conside ed a model o he e olu ion o landscape subjec o wa e e osion, in o de o s udy he o ma ion o pa e ns. This model akes in o accoun he wa e low, he dissol ed sedimen s and he main e ec s o e osion and sedimen a ion, while emaining simple enough o be s udied bo h heo e ically and nume ically. We p o ed ha unde ealis ic hypo heses, he sys em is well posed locally in ime. Then a comple e spec al analysis o a linea iza ion o he sys em a ound s a iona y solu ions has been pe o med. This analysis highligh s he a ious beha io s o he sys em depending on he pa ame e s and on he equencies o he pe u ba ions. A e y impo an pa ame e o he sys em is he cons an o c eep K, which con ols he s abili y. The sys em can become s able i his cons an is la ge enough and is uns able i his cons an is oo small. Mo eo e , he ins abili ies in he sys em appea in he ans e se di ec ion, and his leads o he o ma ion o channels pa allel o he wa e low. This analysis is a p elimina y s ep in he s udy o pa e n o ma ion on e odible su aces. We ha e shown ha an ins abili y mechanism can explain he o ma ion o pa allel channels a he ea ly s ages o e osion. We plan o pe o m a mo e comple e pa ame ic s udy o he sys em, in o de o unde s and he a ious beha io s o he model. This analysis should be comple ed by nume ical simula ion o illus a e hese beha io s. The e a e se e al in e es ing di ec ion o esea ch. Fi s , he analysis o his model could be ex ended o mo e gene al landscapes like moun ains o alley. In hese cases, he well-posedness o he sys em in sho ime can be s ill alid, as long as he wa e le el and he wa e speed does no anish. A simila s abili y analysis could 123 Page 39 o 55 20 be ca ied ou i s eady s a es exis . One could also conside mo e in ol ed models. Indeed a lo o ac o s a e no aken in o accoun as wea he , ege a ion, animal and human ac i i ies. I is be qui e di icul o include hese ac o s in he model, bu a possible amelio a ion would be o include andomness in he equa ions. A andom e m could allow o model hese ac o s which luc ua e o e ime. Mo eo e , a non- cons an sou ce e m ha would model he a ia ions o ain in ime would make he model mo e ealis ic. Ano he po en ial imp o emen o he model would be o conside mo e complex laws o he luid eloci y o e en conside shallow wa e - ype models o he e olu ion o he laye o luids. Acknowledgemen s PD holds a isi ing p o esso associa ion wi h he Depa men o Ma hema ics, Impe- ial College London. A ailabili y o Da a and Ma e ials The code o nume ical simula ions gene a ed du ing he cu en s udy is a ailable a h ps://gi hub.com/juliebina d/landscape_e ol_ ini e_ olum. Decla a ions Con lic o in e es No unding was ecei ed o assis wi h he p epa a ion o his manusc ip . A P oo o he S abili y Theo em This sec ion is de o ed o he p oo o Theo em 4.1. Fi s we s a e he Rou h–Hu wi z c i e ia o complex polynomial o deg ee 3, p o ed in Mo is (1962): P oposi ion A.1 Le P(X)=X3+(a1+ib1)X2+(a2+ib2)X+a3+ib3be a polynomial wi h complex coe icien s. Then he oo s o P ha e a posi i e imagina y pa i and only i he h ee ollowing condi ions a e sa is ied: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ −2=−1a1 0b1>0 4= 1a1a2a3 0b1b2b3 01 a1a2 00 b1b2  >0,−6=−  1a1a2a300 0b1b2b300 01 a1a2a30 00 b1b2b30 00 1 a1a2a3 00 0 b1b2b3  >0 This p oposi ion is di ec ly used o p o e Theo em 4.1. 20 Page 40 o 55 P oo o Theo em 4.1 The cha ac e is ic polynomial o A, deno ed by P,is: P(X)= X+iξ an θ+α2h0α2h −amc h+αiξanc an θX+a+iξ an θαiξanc an θ amc ¯ρs−αiξanhc ¯ρs an θ−ah ¯ρs X−αiξanhc ¯ρs an θ+2K  = X+iξ an θ+α2h0α2h −amc h+αiξanc an θX+a+iξ an θαiξanc an θ 0hX+iξh an θ ¯ρs X+2K  . whe e we ha e deno ed 2=ξ2+η2. In o de o simpli y he compu a ions, we in oduce he a iables Y:= X+iξ an θ, ¯ ξ=ξ an θ, ¯2=αh2,¯ Kαh=K,¯ρs¯ h=h. We also de ine he cons an s N=αanc/ an2θ,M=amc/h. Consequen ly, he polynomial w i es as: P(Y)= Y+¯20¯2 −M+iξNY+ai ¯ ξN 0¯ hY Y +¯2K−i¯ ξ =−¯ hY i¯ ξNY +¯2M+(Y+a)(Y+¯2)(Y−i¯ ξ+¯2¯ K) =Y3+"a+¯2(1+¯ K)−i¯ ξ(1+N¯ h)#Y2 +"¯2(a(1+¯ K)−¯ hM +¯ K¯2)−i¯ ξ(a+¯2)#Y +a¯2(¯2¯ K−i¯ ξ). (38) The sys em is s able i and only i he oo s o X→ P(X)ha e a nega i e eal pa . As Yha e he same eal pa as X, his is equi alen o he ac ha he oo s o Y→ P(Y) ha e a nega i e eal pa . Deno ing iλ:= Y, he sys em is s able i and only i he oo s o λ→ P(iλ) ha e a posi i e imagina y pa . Thus, we will apply he Rou h–Hu wi z c i e ia o he polynomial λ→ Q(λ) := iP(iλ) whe e iλ:= Y: Q(λ) =λ3−"¯ ξ(1+N¯ h)+ia+¯2(1+¯ K)#λ2−"¯2(a(1+¯ K)−¯ hM +¯ K¯2) −i¯ ξ(a+¯2)#λ +a¯2(¯ ξ+i¯2¯ K). Page 41 o 55 20 We deno e: ¯a1=1+N¯ h, ¯ b1=a+¯2(1+¯ K), ¯a2=a(1+¯ K)−¯ hM +¯2¯ K, ¯ b2=a+¯2. Thus, Q(λ) =λ3−¯ ξ¯a1+i¯ b1)λ2+−¯2¯a2+i¯ ξ¯ b2λ+a¯2(¯ ξ+i¯2¯ K). The i s condi ion: The de e minan −2w i es as: −2=−1−¯ ξ¯a1 0−¯ b1=¯ b1=a+¯2(1+¯ K). Consequen ly, as a>0 and K≥0, he condi ion −2>0 is always sa is ied. The second condi ion: The de e minan 4is gi en by: 4= −¯ b1¯ ξ¯ b2¯4a¯ K 1−¯ ξ¯a1−¯2¯a2 0−¯ b1¯ ξ¯ b2= −¯ b1¯ ξ(¯ b2−¯a1¯ b1)¯2(¯2a¯ K−¯ b1¯a2) 10 0 0−¯ b1¯ ξ¯ b2 =¯2¯ ξ2 ¯2¯ b2¯a1¯ b1−¯ b2+¯ b1¯a2¯ b1−¯2a¯ K. Le =¯ ξ2 ¯2∈[0,μ an2θ eh ], hen one inds ha 4>0∀(ξ, η) ∈(R2)∗i and only i ¯ b2¯a1¯ b1−¯ b2+¯ b1¯a2¯ b1−¯2a¯ K>0∀ ∈[0,μ an2θ eh ],∀¯2>0. The e ms ¯ b1and ¯ b2a e posi i e, and: ¯a1¯ b1−¯ b2=a+¯2(1+¯ K)+N¯ h(a+¯21+¯ K)−a−¯2 =¯2¯ K+N¯ ha+¯2(1+¯ K)>0. Consequen ly, one has 4>0,∀(ξ, η) ∈(R2)∗i and only i ¯a2¯ b1−¯2a¯ K≥0∀¯2>0. We compu e: ¯a2¯ b1−¯2a¯ K=a(1+¯ K)−¯ hM +¯2¯ Ka+¯2(1+¯ K)−¯2a¯ K =aa(1+¯ K)−¯ hM+¯2(1+¯ K)a(1+¯ K)−¯ hM +¯2¯ K. 20 Page 48 o 55 We can suppose, up o eno maliza ion ha V(ξ, η) =O +∞(1). We calcula e a Taylo expansion o λ(ξ, η) ξ2+η2=0+1(ξ, η) +2(ξ, η) +o(1), and o V(ξ, η) =V0+V1(ξ, η) +V2(ξ, η) +o(1 ξ2+η2). O de 0: When ξ2+η2→+∞, Sys em (43) becomes A2V0=−0V0. I s solu ions a e he eigen alues and eigen ec o s o −A2,so1 0=−K,2 0=−αhand 3 0=0. The associa ed eigen ec o s a e X1=⎡ ⎣ αh 0 K−αh⎤ ⎦,X2=⎡ ⎣ 1 0 0⎤ ⎦,X3=⎡ ⎣ 0 1 0⎤ ⎦. When K>0, he e is one eigen alue bi u ca ing om ze o, 3, and wo o he eigen alues ha e nega i e eal pa s. When K=0, he wo eigen alues 1and 3 a e bi u ca ing om ze o. Consequen ly, we will s udy hese wo cases. C.1TheCaseK>0 In his pa , we con inue he asymp o ic expansion o he eigen alue 3. The ze o-o de e ms o he eigen ec o s a e V1 0=X1,V2 0=X2and V3 0=X3. We calcula e he i s -o de e m o 3. The i s -o de e ms o Sys em (43) when ξ2+η2→+∞depends on he asymp o ic beha io o iξ.I ξ→+∞, hen he i s -o de e ms o he sys em a e: A2V3 1−iξA1 ξ2+η2V3 0=−3 1V3 0.(44) And i ξis bounded, iξA1and A0a e o he same o de ; hus, he i s -o de e ms a e: A2V3 1−iξA1+A0 ξ2+η2V3 0=−3 1V3 0.(45) Jou nal o Nonlinea Science (2024) 34 :20 Page 49 o 55 20 To compu e 3 1, we mul iply he equa ions by he le eigen ec o o A2associa ed wi h he eigen alue 0, X∗=X3. Then, a e some compu a ions: I ξ→+∞, 3 1=−iξ an θ ξ2+η2,V3 1=1 ξ⎡ ⎣ 0 1 0⎤ ⎦. I ξbounded, 3 1=−iξ an θ ξ2+η2−sρs ehcsa 1 ξ2+η2. When ξis bounded, we al eady compu ed he second-o de e m o 3. When ξ→+∞, he second-o de e ms o he sys em a e: A2V3 2−iξA1 ξ2+η2V3 1−A0 ξ2+η2V3 0=−3 1V3 1−3 2V3 0. Then, one inds: 3 2=− sρs ehcsa 1 ξ2+η2. Finally, he asymp o ic expansion o he eigen alues o A(ξ, η) when ξ2+η2→ +∞and K>0is: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ λ1(ξ, η) =−K(ξ2+η2)+o(ξ2+η2) λ2(ξ, η) =−αh(ξ2+η2)+o(ξ2+η2) λ3(ξ, η) =−iξ an θ−s¯ρs eh +o(1) C.2TheCaseK=0 When K=0, he ze o-o de e ms o he eigen alues 1and 3 anish. Thus, in o de o p o e he second poin o P oposi ion 4.2, we need o compu e he nex o de e ms o hese wo eigen alues. The eigen ec o s associa ed wi h he wo ze o eigen alues o −A2a e X1=⎡ ⎣ αh 0 −αh⎤ ⎦,X3=⎡ ⎣ 0 1 0⎤ ⎦. Thus, he e exis cons an s u1, 1,u3, 3such ha V1 0=u1X1+ 1X3and V3 0= u3X1+ 3X3. The le eigen ec o s associa ed wi h he ze os eigen alues o −Ca e: X1 ∗=001 ,X3 ∗=010 . 20 Page 50 o 55 I ξ→+∞, hen he i s -o de e ms o Sys em (43)a e: A2V3 1−iξA1 ξ2+η2V3 0=−3 1V3 0.(46) I ξ= 0 is ixed, he i s -o de e ms a e: A2V3 1−iξA1+A0 ξ2+η2V3 0=−3 1V3 0.(47) To compu e 1 1and 3 1, we mul iply he equa ions (46) and (47)byX1 ∗and X3 ∗. Then, when ξ→+∞, one has: ⎧ ⎨ ⎩ 1 1=0,V1 0=X1, 3 1=−iξ an θ ξ2+η2,V3 0=X3. And when ξ= 0 is ixed, ⎧ ⎪ ⎨ ⎪ ⎩ 1 1=scm eh , 3 1=−iξ an θ+s¯ρs eh 1 ξ2+η2. When ξ→+∞, he second-o de e ms o Sys em (43)a e: A2V2−iξA1 ξ2+η2V1−A0 ξ2+η2V0=−1V1−2V0. Then, one ob ains: 1 2=sc eh m−ξ2 2n, 3 2=−s¯ρs eh 1 ξ2+η2. As a conclusion, he asymp o ic expansion o he eigen alues o A(ξ, η), when ξ2+η2→+∞and ξ= 0, K=0is: ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ λ1(ξ, η) =sc eh m−ξ2 2n+o(1), λ2(ξ, η) =−αh(ξ2+η2)+o(ξ2+η2), λ3(ξ, η) =−iξ an θ−s¯ρs eh +o(1). Finally, when ξ=0 he cha ac e is ic polynomial Pcan be ac o ized: P(X)=XX2+(a+α2h)X+α2ah 1−mc ¯ρs. Page 51 o 55 20 Thus, i has a ze o oo , λ1=0, and he wo o he oo s a e λ2,3=−α2h−a±(a−α2h)2+4α2amhc ¯ρs 2. Consequen ly, he asymp o ic de elopmen o he eigen alues o he sys em when η2→+∞and ξ=0, K=0is ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ λ1(0,η)=0, λ2(0,η)=−αhη2+o(η2), λ3(0,η)=s¯ρs eh mc ¯ρs−1+o(1). This achie es he p oo o P oposi ion 4.2. D Nume ical Scheme In his appendix, we gi e he nume ical scheme used o he simula ions o he sys em. S a iona y egime o he wa e The a io be ween he e osion speed and he luid eloci y is small and we will suppose ha he ime de i a i es o cand ha e small. In o de o jus i y his assump ion, we escale he equa ions by in oducing he cha ac e is ic a iables Z he e oded heigh , L he cha ac e is ic leng h and T=Z/e he cha ac e is ic ime. We de ine he dimensionless a iables: h:= h H,z=z Z, := V,c:= c csa , x:= x L, := e Z . The e o e, d opping he p imes, Sys em (8a) is w i en as: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ h+1 di (h ) = e, ∂ (ch)+1 di (ch ) =ρs csa hm| |n−s ec, ∂ z=$ Kz−hm| |n+s ec, =−μ V∇(h+z). He e, we de ined := Le HV .I 1, as gene ally ρscsa , we can neglec he e ms ∂ hand ∂ (ch)in he equa ion he ch. Thus, assuming enough egula i y on he solu ions, we can w i e di (ch ) =h .∇c+cdi (h ) =h .∇c. Mo eo e , we ix 20 Page 52 o 55 V=μ o simplici y. We ob ain: ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ di (h ) =$ , h .∇c=ρs csa hm| |n−s ec, ∂ z=$ Kz−hm| |n+s ec, =−∇(h+z). (48) Disc e iza ion o he equa ions The cons ain on ha each ime di (h∇(h+z)) =0 leads o a ully nonlinea p oblem ha may be ha d o sol e nume ically. Ins ead, we disc e ize his equa ion as di (hn−1∇hn)+di (hn∇zn)=− ,(49) whe e hn ep esen s he luid heigh a ime n=nδ , whe eas he su ace heigh a ime n,zn, has been calcula ed by an explici Eule me hod, using he solu ions a ime n−1, hn−1and cn−1:∀1≤i≤Nx,1≤j≤Ny, zn i,j−zn−1 i,j d =K zn−1 i+1,j−2zn−1 i−1,j+zn−1 i,j dx +K zn−1 i,j+1−2zn−1 i,j−1+zn−1 i,j dy −(E−S)n i,j. We disc e ize he equa ion on hnby a cen e ed ini e olume scheme, which w i es: − i,j=hn−1 i+1/2,j(hn i+1,j−hn i,j)−hn−1 i−1/2,j(hn i,j−hn i−1,j) dx2 +hn−1 i,j+1/2(hn i,j+1−hn i,j)−hn−1 i,j−1/2(hn i,j−hn i,j−1) dy2 +hn i+1/2,j(zn i+1,j−zn i,j)−hn i−1/2,j(zn i,j−zn i−1,j) dx2 +hn i,j+1/2(zn i,j+1−zn i,j)−hn i,j−1/2(zn i,j−zn i,j−1) dy2, whe e hn−1 i+1/2,j=hn−1 i,j+hn−1 i+1,j 2. Finally, Equa ion (37)oncis disc e ized by an upwind ini e olume scheme, conside ing he a iable xas a ime a iable. The ime dis- c e iza ion is an explici Eule scheme. ci+1,j−ci,j d +max( x[i,j],0) y[i,j] ci,j−ci,j−1 dy +min( x[i,j],0) y[i,j] ci,j−ci,j+1 dy =− ρs hi,j x[i,j](E−S)n i,j. Page 53 o 55 20 Fig. 17 E oded heigh o he soil in mm, o K=Ke,a T=2s Bounda y condi ions We choose pe iodic bounda y condi ions in he ydi ec ion. The e emains wo bounda ies: he op and he bo om o he il ed plane. The bounda y condi ions o he soil heigh a e Neumann condi ions: ∀1≤j≤Ny, zn 0,j=zn 1,j,zn Nx,j=zn Nx+1,j. The bounda y condi ions o he wa e heigh and o he concen a ion o sedimen s a e a Di ichle condi ion a he op because he incoming low is ixed and a ee low Neumann condi ion a he bo om. hn 0,j=h0,hn Nx,j=hn Nx+1,j, cn 0,j=c0,cn Nx,j=cn Nx+1,j. Compa ison wi h a scheme ha sol es he non-s a iona y sys em In his pa ag aph, we d op he s a iona y assump ion o he wa e heigh and sedi- men concen a ion, and compa e he esul s o hose whe e he s a iona y assump ion is made. We sol e Sys em (8a), wi h a ini e olume scheme. A ully explici scheme ails o sol e he sys em, because he compu ed solu ion quickly blows up, e en wi h a ime s ep smalle han he one gi en by he CFL condi ion: δ ≤δx wa e . The e o e, we use a semi-explici scheme: Linea e ms o he wa e heigh and sed- imen concen a ion equa ions a e implici , and we explici a pa o he nonlinea e ms. The disc e iza ion in ime o equa ions on hand cis gi en by: ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ hk+1−hk δ +μdi (hh∇hk+1+hk+1∇zk+1)=0, hk+1ck+1−ck δ +hk+1 k+1.∇ck+1=ehk+1m k+1n−ρssck+1 csa , k=−μ∇(hk+zk). (50) 20 Page 54 o 55 Disc e iza ion in space is done by a ini e olume scheme, he same as o he s a iona y Sys em (48). The nume ical pa ame e s a e gi en by Table 1, and δ =0.1s. 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