Abstract
The modelling of time-varying shallow flows, such as tides and storm surges, is complicated by the nonlinear dependency of bed shear stress on flow speed. For tidal flows, Lorentz’s linearisation circumvents nonlinearity by specifying a (steady) friction coefficient r based on a tide-averaged criterion of energy equivalence. However, this approach is not suitable for phenomena with episodic and irregular forcings such as storm surges. Here, we studied the implications of applying Lorentz’s energy criterion in an instantaneous sense, so that an unsteady friction coefficient r(t) adjusts to the temporal development of natural wind-driven flows. This new bed-stress parametrisation was implemented in an idealised model of a single channel, forced by time-varying signals of wind stress (acting over the entire domain) and surface elevation (at the channel mouth). The solution method combines analytical solutions of the cross-sectionally averaged linearised shallow-water equations, obtained in the frequency domain, with an iterative procedure to determine r(t). Model results, compared with a reference finite-difference solution retaining the quadratic bed shear stress, show that this new approach accurately captures the qualitative and quantitative aspects of the surge dynamics (height and timing of surge peaks, sloshing, friction-induced tide-surge interaction) for both synthetic and realistic wind forcings.
Highlights
Bottom friction is important for shallow water flows, e.g., those associated with tides and storm surges in the marine environment, as discussed in [1]
An a priori scale analysis (Appendix B) shows that, for these parameter values, bottom friction is of first order importance
We devised and tested a novel time-dependent linearisation of bed shear stress extending Lorentz’s classical linearisation to the idealised modelling of storm surges in channels. This approach consists of two elements: (i) the formulation of an instantaneous power equivalence criterion to specify an unsteady channel-averaged friction coefficient r (t); and (ii) a straightforward iterative procedure, carried out in the frequency domain, to obtain both r (t) and the corresponding solution of the linearised shallow-water equations
Summary
Bottom friction is important for shallow water flows, e.g., those associated with tides and storm surges in the marine environment, as discussed in [1]. The nonlinear dependency of bed shear stress on flow speed contributes to various phenomena such as tide-surge interactions [2,3] and the deformation of tidal curves [1]. This nonlinearity may complicate solution techniques in processbased models. This is the case for the subclass of idealised or exploratory process-based models, e.g., [4], which aim to study physical phenomena in isolation. These models usually schematise processes and geometry, enabling one to determine the analytical solution to (parts of) the problem, which in turn allows for rapid extensive sensitivity analyses
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