Deterministic Methods in Systems Hydrology - Chapter 8

Mô hình xác định phi tuyến tính8,1 KHÔNG tuyến tính về thuỷ văn Nếu chúng ta xem xét các phương trình vật lý cơ bản về các quá trình thủy văn khác nhau, chúng ta thấy rằng những phương trình này (và do đó các quá trình mà họ đại diện) là phi tuyến tính. Do đó, chúng ta phải đối mặt với khả năng riêng biệt rằng tất cả các phương pháp tiếp cận phân tích tuyến tính được thảo luận trong chương 4, 5, 6 và 7 có thể không liên quan đến vấn đề thủy văn thực...
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Deterministic Methods in Systems Hydrology - Chapter 8 CHAPTER 8 Non-linear Deterministic Models 8.1 NON-LINEARITY IN HYDROLOGY If we examine the basic physical equations governing the various hydrologic processes, we find that these equations (and hence the processes they represent) are non-linear. Consequently, we face the distinct possibility that all of the approaches of linear analysis discussed in Chapters 4, 5, 6 and 7 may be irrelevant to real hydrologic problems, save as a prelude to the development of non-linear methods. Accordingly, in theNon-linear present chapter we take up this question of non-linearity and ask ourselves whether wemethods can determine under what circumstances the effects of non-linearity will be most marked and also whether we can adapt the methods of linear analysis described in previous chapters to the non-linear case. While knowledge of linear methods of analysis is valuable in such an examination, we must avoid the tendency to carry over into non-linear analysis certain preconceptions, which are valid only for the linear case. The basic equations for the one-dimensional analysis of unsteady flow in open channels are the continuity equation and the equation for the conservation of linear momentum. The continuity equation can be written as: Q A (8.1)  r ( x, t )  x t where Q is the discharge, A the area of flow, and r(x, t) the rate of lateral inflow. The above equation is a linear one and consequently poses no difficulties for us in this regard. The second equation used in the one-dimensional analysis of unsteady free- surface flow is that based on the conservation of linear momentum, which reads y u u 1 u u (8.2)  S0  S f  r ( x, t )   x g x g t gy where y is the depth of flow, u is the mean velocity, S0 is the bottom slope and Sf is the friction slope. This dynamic equation is highly nonlinear. Consequently, it is not possible to obtain closed-form solutions for problems governed by equations (8.1) and (8.2). The extent of the non-linearity can be appreciated if we examine the special case of discharge in an infinitely wide channel with Chezy friction, in which case the continuity equation takes the form q y (8.3)  r ( x, t )  x t where q = uy is the where q = u y is the discharge per unit width; and the momentum equation takes the form u2 y u u 1 u u (8.4a)  S0  2  r ( x, t )   x g x g t C y gy which appears to be non-linear in only three of its six terms. However, if we multiply through by g y, f ive of the six terms of the equation are seen to be non-linear. If, in - 143 - addition, we express u in terms of q and y, which are the dependent variables in the linear continuity equation, we obtain y q q g ( gy 3  q 2 )  2qy  y 2  S0 gy 3  2 q 2 (8.4b) x x t C in which every term is seen to be highly non-linear (see Appendix D). On the basis of the above equations we would expect such processes as flood Unsteady flow with a routing, which is a case of unsteady flow with a free surface, to be characterised byfree surface highly non-linear behaviour. However, practically all the classical methods of flood routing commonly used in applied hydrology are linear methods. In contrast most of the methods used in applied hydrology to analyse overland flow (which is another case of unsteady free surface flow) are non-linear in ...