Long-term morphodynamic modelling of tidal basins with rivers

Chapter 4: Linear approximation of ASMITA equations 27 4 Linear approximation of ASMITA equations 4.1 Introduction In this chapter, a simplified approximation of ASMITA equations will be done trying to obtain more insight into the morphological behaviour of tidal inlets when a river discharge is considered. For this, Taylor series are used and the approximations are done around the equilibrium volume of each element. Thanks to these simplifications, we can obtain expressions which let us know how volumes evolve with time. We will start with a tidal system constituted by one element, and increase the difficulty of the system, adding elements one by one, until a system with three elements. Finally, new ASMITA equations will be presented in the case of a n-elements system. 4.2 One element in the tidal system 4.2.1 Model equations First, a system with only one element will be considered. As we also may take into account the river effect, this single element might be the channel, otherwise the river effect would not be reflected properly. We will present a figure where exchange sediment relations between elements is visible. Q Q Q Qw ww w Qc QcQc Qcc cc c Q Q Q Qw ww w () oc c E cc δ − () cc c ce wA c c− outside world S SS S channel wet volume dry volume Figure 4-1 Sediment exchange in case of one element in the system After this report Chapter 4: Linear approximation of ASMITA equations 28 In this case, model equations can be written as it follows: () () cE c sccec cc QcSwAc c δ −+ −= − (4.1) () c s cce c dV wA c c dt =− (4.2) r ce ce E c V cc V æö =ç÷ èø (4.3) From the first equation, which corresponds to the mass-balance equation, we can determine c c: r ce EscE c c sc V cSwAc V cQwA δ δ æö ++ ç÷ èø =++ (4.4) We will substitute c c and ce c into the equation that represents morphological changes (4.2) and we obtain: r ce rEscE c cce sc E csc V cSwAc V dV V wA c dt V Q w A δ δ éù æö ++ êú ç÷ æö êú èø =− ç÷ êú ++ èø êú êú ëû (4.5) Now, the only variable that is unknown is the actual volume of the channel: c V. We will write again the previous equation in another way: () () r csc ce EE sc c dV w A V cQ cS dt Q w A V δδ δ ìü æö ïï =+−+ íý ç÷ ++ èø ïï îþ (4.6) And it can be schematise as it follows: r c c dV a AB C dt V ìü æö ïï =− íý ç÷ èø ïï îþ (4.7) Chapter 4: Linear approximation of ASMITA equations 29 From this, it becomes clear that the equation above is a non-linear differential equation. In order to solve it, we will linearise this equation, using Taylor series approximations, around the equilibrium volume Vce, which corresponds to the constant a in the equation (4.7). The second and higher order terms will be neglected. We will change our variable c V to ' c V, as it follows: ' ccce VVV=− (4.8) And, after linearising equation (4.7) we will obtain a linear nonhomogeneous differential first order equation: () '''' ' c cE c dV rBA ABV AcQS V dt a =− = −− (4.9) Where: () 'sc E s c wA cQ S AQwA δ − =++ (4.10) () '() scE sc rw A c Q BQwAa δ δ + =++ (4.11) Next, the solution of the previous differential equation is presented: '(') t co VAVAe τ ττ − =+ − (4.12) Where : () () ce s c scE VQwA rw A c Q δ τδ ++ =+ (4.13) sc s c wA AQwA δ =++ ; (4.14) () () Ece E cQ SV Arc Q τδ − =+ (4.15) () ' , and 0 ooce o VVV VVt=− = =, this is the initial condition necessary to solve the differential equation Chapter 4: Linear approximation of ASMITA equations 30 From equation (4.12) we can see that for t→∞, a steady state is reached, as ' c V tend to an asymptotic value. The corresponding channel volume is: ,1() E cce E cQ S VV rc Q δ ∞ æö − =+ ⋅ ç÷ + èø (4.16) This volume, henceforth called the end volume, deviates from the equilibrium volume ce V as it is clearly expressed in last equation. We might keep in mind that that ce V corresponds to an inlet system under the influence of a tide only. With the new variable we have introduced, the river, the system needs to be transporting sediment continuously from the channel to the outside world, according to the tendency to reach an equilibrium situation; otherwise the channel would disappear, because of the extra amount of sand accumulated. Despite of that, we might not forget that the river means also a water discharge for the channel. As explained before, the transport exists when the system is not in equilibrium, then, in our case, the channel needs to be all the time in an "out of equilibrium" state so the transport is able to occur constantly. 4.2.2 Time-scales One can interpret a time-scale as the characteristic time for adaptation, once an element is moved from its equilibrium, and it might be different for each element. Also, this time-scale let us know about the first response of an element in front of a disturbance. According to Eysink (1990), one can define the time-scale as it follows (in case of no river in the system): 0 ' ' o t V dV dt τ = =− (4.17) The nominator can be interpreted as the initial volume and the denominator as the initial rate of adaptation. Also, this can be seen graphically as: Chapter 4: Linear approximation of ASMITA equations 31 ti me τ o V ()Vt As one can observe in the previous graph, the time scale τ corresponds to the intersection between the tangent at t=0 and the time axis. Furthermore, we can read the previous equation as: 0 '' o t dV V dt τ = =− (4.18) It can be concluded that as larger is the slope of the tangent line, which means 0 ' t dV dt = bigger, as lower is the value of τ ( ' o V does not change). Also, low values of time scale mean that in less time the system will reach a steady situation. Let's write again the previous equation that describes volume evolution in the channel in case of a river discharge: '(') t co VAVAe τ ττ − =+ − (4.19) We will change A τ to ,' c V∞, thus: ,, ''(' ') t ccoc VV VVe τ − ∞∞ =+− (4.20) Figure 4-2 Time-scales graphical interpretation After this report Chapter 4: Linear approximation of ASMITA equations 32 In this case, just derivating previous expression, the rate of change is given by: , '' 't cc cVV dV e dt τ τ − ∞ − éù =−êú ëû (4.21) According to Eysink (1990), the time scale parameter can be expressed as it follows: , 0 '' ' cc c t VV dV dt τ ∞ = − = (4.22) Notice that last expression can be obtained, too, from equation (4.21) when t=0. We will try to understand the meaning of previous formula. First of all, we will pay attention to the nominator: () () * ,,, '' cc cce c ce cc c VV VV V V VV V ∞∞∞ −=−−−=−= (4.23) Hence, this is the difference between the end volume ,c V∞ and the equilibrium volume ce V. Then, * c V is a relative volume, but compared to the end volume instead of the equilibrium volume. The term τ (the time scale) will be the intersection between the tangent to the curve '( ) c Vt when t=0 and the horizontal straight line , '' cc VV ∞ =. Thus: * * 0 o t V dV dt τ = =− and *'dV dV dt dt = (4.24) because they only differ from a constant which disappear once we derive. Next graphs help to understand previous explanations: Chapter 4: Linear approximation of ASMITA equations 33 '( ) c Vt , 'c V∞ τ time Figure 4-3 Time-scales using volumes relative to Veq After this report *() c Vt ,c V∞ τ time Figure 4-4 Time-scales using volumes relative to Vend After this report In the expression for time scales, in both situations: with and without river, the term τ, the so called time-scale, match up with the term that appears in the power of the exponential function (that describes the element behaviour with time). In the next chapters, when we consider more than one element in the tidal system, we will see that the term τ, defined as in the equation (4.17) and the term that appears in the power of the exponential function are not the same. However, this will be explained more clearly when expressions for a system with more than one element are written. Chapter 4: Linear approximation of ASMITA equations 34 4.2.3 Application of the linearised equations Let's consider an example to visualise the behaviour of the system. We will also compare this situation with a situation without river discharge (keeping the rest of the inputs with the same value). Before that, we will present some clarifications to make clear the name used for the different variables. The term relative volume correspond to 'V, hence, is the volume compared to the equilibrium volume, in other words, it shows how much a volume differs from the equilibrium volume (defined in a no river discharge situation). The term end volume is the volume at the steady situation reached by the system when enough time has passed. It corresponds to the nonhomogeneous term in the equation (4.20), and it is the term A τ in equation (4.19), for instance. Thus, it is the limit when time tend to infinite, and it is equal also to the volume that makes the derivative 'dV dt become zero, as there is an asymptotic character in the volumes evolution. Here we present the data used as an example. These data’s are obtained based on Van Goor (2001) work, and we have included an inexistent river discharge: cE ws n δoc Ac h Vo Q S 0.0002 5e-5 m/s 2 1500m3/s 9.83e7 m3 2 m variable 200 m3/s 0.2 m3/s Table 1 Input data for one element system After Van Goor (2001) According to Martin, J.P (1997), the sediment input is been estimated as approximately 1/1000 x Q. Next, a graph that shows how the volume of the channel change with time in case of a river discharge in the basin is presented, using previous input data: Chapter 4: Linear approximation of ASMITA equations 35 Q=200m3/s, S=0,2m3/s -1,5E+08 -1,0E+08 -5,0E+07 0,0E+00 5,0E+07 1,0E+08 1,5E+08 0 50 100 150 200 250 300 350 t(years) wet relative volume (m3) Vo=2e8 Vo=4e8 Vo=6e8 Vo=1e8 Clearly, one can see the asymptotic tendency of the element since it evolves towards a steady state, which do not correspond to the original equilibrium volume. In the following graph we will see the behaviour of the channel when there is no river discharge. It will evolve to the own equilibrium volume, to a relative volume equal to zero: Q=0, S=0 -1,5E+08 -1,0E+08 -5,0E+07 0,0E+00 5,0E+07 1,0E+08 1,5E+08 2,0E+08 0 50 100 150 200 250 300 350 t(years) wet relative volume (m3) Vo=2e8 Vo=4e8 Vo=6e8 Vo=1e8 Figure 4-5 Channel volume evolution with river discharge After this re p ort Figure 4-6 Channel volume evolution without river discharge After this report Chapter 4: Linear approximation of ASMITA equations 42 The constants C1 and C2 may show us how the elements evolve, related to the (system) time-scales T1 and T2. We shall not forget that these constants depend on the initial conditions and that initial conditions do not influence the end volumes in each element. From next section, where we apply the linearised equations to an example, we take the expression (4.48) and leave C1 and C2, thus: 8 36,1 15,3 12 7 '1 1 1,34 10 '0,38 1,29 3,08 10 tt c d VCeCe V −− æö −⋅ æö æö æö =++ ç÷ ç÷ ç÷ ç÷ −⋅ èø èø èø èø (4.47) To see more clearly which is the influence of each constant, we will give the value equal to the unit to one of them, and observe how the other one evolve when we change its value. In the following graphs we can see first the effect of keeping C1 equal to the unit and changing C2, and second the effect of changing C1 while C2 remains constant and equal to the unit: -1,5E+08 -1,0E+08 -5,0E+07 0,0E+00 5,0E+07 1,0E+08 0 20406080100 t (years) V' (m3) Figure 4-9 Influence of C1 integration constant After Kragtwijk (2001) C1 variable, C2 =1 C CC C1 11 1>0 >0>0 >0 C CC C1 11 1<0 <0<0 <0 C CC C1 11 1<0 <0<0 <0 C CC C1 11 1>0 >0>0 >0 channel delta Chapter 4: Linear approximation of ASMITA equations 43 -1,5E+08 -1,0E+08 -5,0E+07 0,0E+00 5,0E+07 1,0E+08 0 20 40 60 80 100 t (years) V' (m3) From these figures, one can observe that the initial development is determined by the smallest time-scale which correspond to T2, this becomes clear in the first figure. After a while, the response of the smallest time-scale disappears, and what remains is the development of the largest time-scale. One can see that observing the second figure. Also, depending on the sign of the constant, we can see an increasing or decreasing behaviour. 4.3.3 Application of the linearised equations In order to visualise the behaviour of the elements, we will consider some data and used the linearised equation just solved. Next, the data are presented: cE wsi n δij Ai h Voi Q S Channel 0.0002 1e-5 m/s 2 1500m3/s 7.47e7 m22 m 2e7 m3 200 m3/s 0.2 m3/s Delta 0.0002 5e-5 m/s 2 1500m3/s 9.83e7 m2 2 m 5e7 m3 200 m3/s 0.2 m3/s Table 2 Input data for a two elements system After Van Goor (2001) C1 =1, C2 variable Figure 4-10 Influence of C2 integration constant After Kra g twi j k (2001) C CC C2 22 2>0 >0>0 >0 C CC C2 22 2<0 <0<0 <0 C CC C1 11 1<0 <0<0 <0 C CC C1 11 1>0 >0>0 >0 channel delta Chapter 4: Linear approximation of ASMITA equations 44 The results obtained in this example are: 8 77 36,1 15,3 7 '1 1 1,34 10 4,63 10 6,0 10 '0,38 1,29 3,08 10 tt c d Vee V −− æö −⋅ æö æö æö =⋅ +⋅ + ç÷ ç÷ ç÷ ç÷ −⋅ èø èø èø èø (4.48) And the corresponding graph: Volumes -1,5E+08 -1,0E+08 -5,0E+07 0,0E+00 5,0E+07 1,0E+08 0 50 100 150 200 250 300 350 time (years) element volume (m3) channel delta From the graph we confirm the tendency of the elements to evolve to a steady state. As in the case of one element, the system needs to be transporting the sediment from the river to the sea, what means that there will be horizontal transport continuously from the channel to the delta, and from the delta to the outside world boundary. Afterwards, we will observe the behaviour of the tidal inlet without river discharge, with Q=S=0. As one might be expecting, the system evolves to an equilibrium situation where volumes in each element are equal to the equilibrium volume. Next, the graph that corresponds to the same data as the situation below but without the river discharge is presented: Figure 4-11 Element's volume evolution with river discharge After this report Chapter 4: Linear approximation of ASMITA equations 45 Volumes -2,0E+08 -1,5E+08 -1,0E+08 -5,0E+07 0,0E+00 5,0E+07 1,0E+08 0 50 100 150 200 250 300 350 time (years) element volume (m3) channel delta 4.3.4 Initial morphological responses to different disturbances We know that elements tend to evolve to an equilibrium state, they exhibit an asymptotic behaviour. This evolution is not necessarily monotonous. The initial response of the element immediately after a disturbance can be move away from the end volume, but also, the element can overshoot the end volume. In the first case, we talk about an opposite initial response (bump) and in the second case an overshoot (Kragtwijk, 2001). Next figure will helps us to understand these two initial responses: Figure 4-13 Initial response in front of a disturbance After Kragtwijk (2001) a. overshoot b. bump Figure 4-12 Element's evolution without river discharge After this report Chapter 4: Linear approximation of ASMITA equations 46 We will use the linearised equations for two elements to see whether an "overshoot" or a "bump" response will occur. That will depend on the integration constants and on the initial volumes. They both determine the character of the initial morphological behaviour. To understand the different responses there can be, it is easier if we work with the initial relative volumes, but relative to the end volumes instead of the equilibrium volumes. Anyway, we will also name them initial relative volumes, but we shall not forget they are compared to the end volumes. We will study the sign of these volumes (when t=0) for the example presented before, in chapter 3. Figures showing this study will be presented in Appendix 1. Furthermore, in our example, we should not forget that C1 corresponds to the behaviour at long term and C2 corresponds to the behaviour at short term (compared to the other one), as the time-scales show, according to explanations in section 4.3.2. The procedure followed is presented next. First of all, we rewrite the equation (4.47) at t=0: 8 12 7 '1 1 1,34 10 '0,381,29 3,08 10 co do VCC V æö −⋅ æö æöæö =++ ç÷ ç÷ ç÷ç÷ ç÷ −⋅ èøèø èø èø (4.49) where ''(0) ''(0) co c do d VVt VVt == == (4.50) And we will work with initial volumes compared to the end volumes, hence, with relative initial volumes. Thus: *8 *7 '''1,3410 '''3,0810 co co cend co do do dend do VVV V VVV V =− =+ ⋅ =− =− ⋅ (4.51) Afterwards we will obtain C1 and C2 as a function of the relative initial volumes as it follows: ** 1 ** 2 1, 29 1, 67 0,38 1, 67 co do co do VV C VV C − = + = (4.52) Chapter 4: Linear approximation of ASMITA equations 47 Moreover, it is easy to see that the initial relative volumes are volumes compared to the end volume, which at the same time is also a relative volume, since it is compared to the equilibrium volume. But, also, it is easy to see that the initial relative volumes are the difference between the initial net volume and the end net volume: ()( ) *'' io io iend io ieq iend ieq io iend VVV VV V V VV=− = − − − =− (4.53) Therefore, we can talk either about differences between initial and end volumes; whether they are relative or net volumes do not affect the relative initial volumes * io V. Knowing that the response of the system may change depending on the sign of C1 and C2 (as it has been explained in previous section), we will see for which values of the initial relative volumes those constants are positive or negative. We will consider the plane () ** , co do VV and draw on it the straight lines C1 =0 and C2 =0. Axes * co V and * do V divide the plane in 4 zones with the 4 possible combinations of signs. The same will occur with the constants C1 and C2. Considering both parameters simultaneously (integration constants and initial relative volumes) we obtain the plane divided in 8 zones: -16 0 16 -10 0 10 Vco* Vdo* C1=0 C2=0 1 2 3 4 5 6 7 8 Figure 4-14 Eight zones depending on initial conditions After this report As one can observe, we obtain 8 different areas, depending on the sign of C1 and C2 and on the sign of the initial relative volumes. The sign of the volumes shows whether the initial volume of the element is higher or Chapter 4: Linear approximation of ASMITA equations 48 lower than the end volume. Also, the sign of the constant C2 shows the response of the element in a short term and the sign of the constant C1 shows the response at long term (as just explained previously). Considering all possible combinations between signs of C1 and C2 and initial relative volumes of the channel and the delta, we can determine the character of the initial morphological behaviour. In order to keep in mind the influence of the constants to the behaviour of the system, we will present a figure for the channel and for the delta: channel t (years) volume (m3) c1>0 c1<0 c2>0 c2<0 Figure 4-15 Initial condition constant for the channel After this report delta t (years) volume (m3) c1>0 c1<0 c2>0 c2<0 Figure 4-16 Initial condition constant for the delta After this report Chapter 4: Linear approximation of ASMITA equations 49 Let's consider a case where the delta has a positive initial volume, that is to say an initial volume larger than the end volume. Let's consider C2 positive, then, there is an initial tendency to decrease in the element (this becomes clear if we look at the figure above, number 4-16. Let's consider also that C1 is positive, then, at long term, there will be a tendency to increase. Therefore, we will see that the element overshoots the end volume at a short term (decreasing tendency), and then, tends to evolve to this end volume at long term (the volume is increasing). Considering all possible combinations, we have elaborated a table that summarise these results, and in appendix 1 graphs for each case are included: Situation Disturbance Constants Channel Delta 1 Vc*>0, Vd*>0 C1>0, C2>0 overshoot monotone 2 Vc*>0, Vd*>0 C1<0, C2>0 overshoot monotone 3 Vc*<0, Vd*>0 C1<0, C2>0 monotone monotone 4 Vc*<0, Vd*>0 C1<0, C2<0 monotone mon/bump 5 Vc*<0, Vd*<0 C1<0, C2<0 monotone mon/oversh 6 Vc*<0, Vd*<0 C1>0, C2<0 overshoot monotone 7 Vc*>0, Vd*<0 C1>0, C2<0 monotone monotone 8 Vc*>0, Vd*<0 C1>0, C2>0 monotone mon/oversh Table 3 Initial response of each element After this report 4.4 Three elements in the tidal system 4.4.1 Model equations In the case of three elements, thus channel, flat and delta, an approximation is also done linearising the ASMITA equations. In order to simplify this process, equations have been generalised as it will be shown next. However, to keep in mind equilibrium relations between the elements, we will write down those equations and we will present figure 3-4 again: Chapter 4: Linear approximation of ASMITA equations 50 ()() ( ) od d E dc d c c d sd d de d cc cc QcQcwAc c δδ −+ −− + = − (4.54) () () ( ) cf c f dc c d c sc c ce c cc cc QcSwAc c δδ −+ −+ −= − (4.55) () ( ) cf f c sf f fe f cc wAcc δ −= − (4.56) Q QQ Qw ww w () dc c d cc δ − S SS S Q QQ Qw ww wc cc cc cc c Q QQ Qw ww wc cc cd dd d Q QQ Qw ww w Q QQ Qw ww w () od d E cc δ − () cf f c cc δ − () ff f fe wA c c− () cc c ce wA c c− () dd d de wA c c− dry volume wet volume flat outside world delta channel Figure 3-4 Exchange lines in case of river discharge After this report First of all we will present the general expression for a system with n elements, and afterwards, apply it to our case, the three elements system. ()()  () ,1 1 ,1 1 n nn nn nn nn snnnen cc cc a wAc c δδ ++−− −+ −+= − (4.57) () nn sn n ne n n dV r wA c c dt r =− (4.58) n r ne ne E n V cc V æö =ç÷ èø (4.59) 1,nN= ; and 2 n r=± , “+” for wet volumes, “-“ for dry volumes j is the element where the river discharge goes to N is the total number of elements in the system Chapter 4: Linear approximation of ASMITA equations 51     1 1,1 0, , ( ), 1 ( ) , n nn nnn nnn nnE anj acQS nj aQcc j nN aQcc c nN δ − −+ =< =− = =− +<< =−− = (4.60) where ,1nn δ +, in case nN=, corresponds to , N outside world No δδ = Using equations (4.58) and (4.59), we obtain the general expression for n c: 1, where n r ne n n nE n snn nn n VdV r cc awA Vadt r æö =− = ç÷ èø (4.61) And, substituting this equation to the first one we obtain a differential equation where there are no input parameters unknown: ()  1 1 ,1 ,1 ,1 ,1 11 11 11 ,1 ,1 ,1 ,1 11 nnn nn nn nn nn nnnn nnnnn rrr ne ne ne n nn E nn nn E nn E nnn dV r dV dV adt a ardtadt VVV ccca VVV δδδ δ δδδδ + − ++− − +− +− +− −+−− +− éù +− − − + = êú ëû æö æö æö =−++− ç÷ ç÷ ç÷ èø èø èø (4.62) Let's write the previous expression again in the three elements case, and for each element: 21 21 12 2 1 1 12 12 12 21121 1 (flat) 0 rr ee EE n VV dV dV r cc adt dt r a V V δδ δδ = éù æö æö +−−− + = êú ç÷ ç÷ èø èø ëû (4.63) () 32 12 23 3 23 3 2 2212211 23 23 21 322213 2 122 21 122 2 (channel) rr ee EE rr ee EE n dV V V dV r dV cc adt dt a a r adt V V VV dVQ cQc S VVadt δδ δδ δδδ δ = éù æö æö + −−− + − + + − êú ç÷ ç÷ èø èø ëû æö æö −+−= ç÷ ç÷ èø èø (4.64) Chapter 4: Linear approximation of ASMITA equations 58 become meaningful in the case of the channel. Also, we can see the volumes seem to be constant for high discharge values, hence, it seems there would be no maximum river discharge. The linearised equations are valid for an interval around an equilibrium state where there is no river. Therefore, as larger is the river discharge, as much further we are from this equilibrium volumes, and the first order equations have less validity. In order to better compare last two figures, we present a graph with both results at the same time, only for the channel volume evolution: -0,5 0 0,5 1 1,5 2 2,5 3 3,5 0 100 200 300 400 500 600 700 Q (m3/s) net channel volumes (m3) channel non lin channel lin Also, for high discharges, the linearised equations do not give enough precision since they are a first order approximation of second order differential equations made around a certain point. And, the approximation is good within an interval not too wide around this point. Then, in case of too high discharges, this is not a good approximation because volumes are too far from the initial equilibrium situation, from the point around which we approximate. In this case, linearisation should be made around another point: the end volume, since it is the value around which elements tend to evolve. Furthermore, a relative error (ε) can be calculated as we are able to obtain the “exact” end volumes, that is without a simplification, according to the following expression: ,, , end lin end full end full VV V ε − = (4.80) Figure 4-21 Comparing full equations with linearised equations results After this report Chapter 4: Linear approximation of ASMITA equations 59 where Vend,lin corresponds to the end volume obtained with the linearised equations (the approximate volume) and the term Vend,full corresponds to the end volume in case we use the full equations. Next, we will present a graph where one can see how the relative error increase, specially for the channel volume, when the discharge also increases: 0 0,2 0,4 0,6 0,8 1 1,2 0 100 200 300 400 500 600 700 Q (m3/s) Relative error flat channel delta Figure 4-22 Relative error depending on the river discharge After this report We might notice that Vi,eq=Vi,end when there is no river discharge (Q=0). It becomes clear that the discharge is what makes the system be moved from the original equilibrium situation. There might be a balance between the value of the river discharge and the precision in solving the linearised ASMITA equations. As much larger is the river discharge, as much worth it is to take into account this boundary condition in the model formulation to be closer to reality, but at the same time, the solution has less exactitude. A compromise between both situations might exist. However, expressions for the end volumes with in case of full equations are easy to obtain (they will be presented in next chapter), no matter how large is the river discharge. This becomes a useful tool in having an idea about the error made in the general use of the linearised equations.