Generalizations of the Wedderburn number: Parameterizing upwelling in stratified lakes

Abstract

The effect of lake geometry on wind-induced upwelling, in a two-layer stratified lake with a variable bottom slope and generic planar shape is investigated. (1) The traditional linearized classification parameter for upwelling, the Wedderburn number, is extended to include finite amplitude effects in a rectangular basin; this extension is important when the Wedderburn number is , , 3. (2) The Wedderburn number is generalized to incorporate the influence of a bottom slope and a variable-width basin for both the linear (large Wedderburn number) and nonlinear (small Wedderburn number) assumptions. Laboratory and numerical experiments of a two-layer stratification were performed and used to validate these extensions. (3) Both extensions are combined to derive a methodology to estimate the interface displacement in real lakes where both geometry effects may act together, and this methodology is used to discuss the response of the thermocline to the onset of wind in Lake Como and Mundaring Weir reservoir. In most cases, symmetry of the basin geometry along the fetch is the key characteristic that defines tilting of the interface for real lakes. Asymmetries in terms of bottom slopes induce shortening or stretching of the effective internal fetch, while a larger excursion of the interface results in narrower upwind areas of the lake due to horizontal redistribution of displaced water volumes. This paper gives the correct definition of the length scales to define the Wedderburn number for any stratified basin.

The importance of nonlinear internal waves in a deep subalpine lake: Lake Iseo, Italy

“…The main parameters characterizing the system during the two storms are given in Table 3. The value of W 21 between 0.45 and 0.91 (see Table 3) was such that the theory of Shintani et al (2010) suggests upwelling; that of Horn et al (2001) implies that both wind event responses fall into regime 2, with the basin-scale waves degenerating into solitary waves by nonlinear steepening.…”

“…The degeneration of weakly nonlinear, basin-scale Kelvin and Poincaré waves has recently been investigated by de la Fuente et al (2008Fuente et al ( , 2010 using a weakly nonhydrostatic model. These authors found Kelvin waves steepened, shedding solitary-type waves similarly to the nonrotating case, whereas Poincaré waves lost their symmetry reversibly as nonlinearities increased, exchanging energy with submodes.…”

LAKE 2.0: a model for temperature, methane, carbon dioxide and oxygen dynamics in lakes

“…The situation changes when the thermocline tilt becomes significant, i.e. when the thin interface between epilimnion and hypolimnion reaches the lake surface at its margins (Shintani et al, 2010). In this case it is the term ∂(w f )/∂z that accounts for the eventual lake overturn, i.e.…”

“…complete vertical homogenization of a water body. This process cannot be simulated by 1-D lake models explicitly, but may be diagnosed using Wedderburn (Shintani et al, 2010) and Lake numbers (Imberger and Patterson, 1989). Here, when applying the lake model for the lake under study, we will use Wed-derburn number time series to check the validity of dropping out the "vertical circulation term" 1 .…”

“…This stratification is a key dynamic factor to which other ones have to be compared. Specifically, the wind force impact, disturbing the lake's layered structure, is assessed via the Wedderburn number (W ) (Shintani et al, 2010) or lake number (Imberger and Patterson, 1989), while the significance of Coriolis force can be quantified by comparing Rossby deformation radius (L R ) to a lake's size (Patterson et al, 1984). The two parameters, W and L R , are plotted in Figs.…”