Application Of The Nonlinear Forcing Singular Vector Method In Stability Analysis Of The Lake Eutrophication

The Lake eutrophication problem has been one of most serious environmental problems which is concerned by all the world The lake eutrophic ecosystem is a open and complex nonlinear system In this paperbased on the dynamical model for nutrient cycling in shallow lakesafter the use of conditional nonlinear optimal perturbation(CNOP) is adopted to investgate the instability and sensitivity of the lake ecosystem to finite-amplitude perturbationA new approach of nonlinear forcing singular vector(NFSV) is adopted to investigate the sensitivity of the lake ecosystem to model errorswhich is derived from the uncertainties in the parameterization of subgrid-scal physical processesexternal forcingand numerical discrete schemesetcThe main researches of this paper are concluded as follows:(1) The impact of the nutrient loading rate c which is combined with an constant tendency perturbations on this lake ecosystem has been studied in this paperWhen the outside force f is zerothe bifurcation diagram for the lake eutrophication model with control parameter has two bifurcation points1 c and2c For c less than the bifurcation point1 c or c lager than the bifurcation point2 c whether it is OES or EES is nonlinearly stableIn case of c between1 c and2c the lake ecosystem is nonlinearly unstablewhich indicates that a large enough initial finite-amplitude perturbation can induce a transition from OES(EES) to EES(OES)When f is greater than zerothe bifurcation interval will appear in advance by the change of c.(2) Through the numerical simulationwe can get the constant tendency perturbations that make the ecosystem the largest development and changes by the given constraintsThe error is the NFSVThe reslts show that the effect of small NFSV tendency error on ecosystemthe nutritional status does not has big changeThe effect of big NFSV tendency error on ecosystemsteady state conversion nutritional status is easy to occur.(3) Through the numerical simulationwe can get the stype function type constant tendency perturbations that make the ecosystem the largest development and changes with the given constraintsThe results indicate that the effect of NFSV tendency error depends on time on corresponding to the ifurcationoutside the range of nutritional statusAt the end of perturbation timethe ecosystem steady transition will not occurButcorresponding to the ifurcation inside the range of nutritional statussteady state conversion nutritional status is asy to occur.This results give us a good reference on lake pollution forecast,meanwhileit will provide a good basis for pollution measures.