ZAMM · Z. Angew. Math. Mech. 85, No. 7, 490 – 498 (2005) / DOI 10.1002/zamm.200210187 Analysis and improvement of a perturbation solution for hydraulic ﬂow in a rotating parabolic channel Janek Laanearu 1, ∗ and Peter Lundberg 1 Department of Mechanics, Tallinn University of Technology, Ehitajate tee 5, 19086 Tallinn, Estonia 2 Department of Meteorology/Physical Oceanography, Stockholm University, 10691 Stockholm, Sweden Received 15 February 2002, revised 16 October 2003, accepted 1 September 2004 Published online 25 May 2005 Key words rotating hydraulics, power series, singularities MSC (2000) 76U05 Power series solutions are presented for a problem within rotating hydraulics. A variety of techniques has been used to investigate and improve the convergence of the expansions. The locations of the dominating singularities affecting the convergence properties of the perturbative series were determined using the Domb-Sykes ratio test as well as a Pad´ e- approximant method. Hereafter conformal mapping and series reversion have been applied to extend the parametric range of utility of the series and to accelerate their convergence. c  2005 WILEY-VCHVerlag GmbH & Co. KGaA, Weinheim 1 Introduction The rotating hydraulic framework to be employed uses the stationary, inviscid shallow-water equations, cf. Gill [6]. The passages dealt with are assumed to have cross-sections varying slowly in the along-stream direction, yielding a two- dimensional problem for a geostrophically balanced along-channel ﬂow. The uniform potential vorticity of the ﬂow is speciﬁed by the ratio of the planetary vorticity to the upstream basin depth (taken to equal the potential depth, cf. Boren¨ as and Pratt [4]). The energy is speciﬁed as a Bernoulli function, conserved along transport streamlines. For a parabolic cross-channel geometry h(x)= ˜ β - ˜ αx 2 , the hydraulic problem can be resolved (Boren¨ as and Lund- berg [2]) by calculating the cross-channel points where the water depth vanishes; -a ∗ and b ∗ , viz. the solutions to a ∗2 = 1 ˆ D ∞ r 2  ˆ D ∞ - Δ  (2 + r) - 1 ˆ D 2 ∞ r 2  ˆ ψ i +1/2  (2 + r) - (2 + r) 2 tanh 2  a ∗ + b ∗ 2  +2a ∗ tanh  a ∗ + b ∗ 2  , (1.1) b ∗2 = 1 ˆ D ∞ r 2  ˆ D ∞ - Δ  (2 + r) - 1 ˆ D 2 ∞ r 2  ˆ ψ i - 1/2  (2 + r) - (2 + r) 2 tanh 2  a ∗ + b ∗ 2  +2b ∗ tanh  a ∗ + b ∗ 2  . (1.2) These equations depend on four nondimensional parameters, where r is a composite measure of the rotation rate and channel width (proportional to ˜ α -1 ), Δ corresponds to the threshold height ( ˜ β sill - ˜ β), and ˆ D ∞ is the equivalently scaled potential depth. The ﬂow distribution between the boundary layers of the upstream reservoir is speciﬁed by ˆ ψ i . The solutions are most conveniently represented by the dimensionless cross-sectional area of the ﬂow A = r/( ˆ D ∞ (b ∗ - a ∗ )). If, for a ﬁxed ˆ D ∞ , modest sill heights and a conserved ﬂux, subcritical as well as supercritical so- lutions exist, these correspond to large and small A, respectively. For increasing sill heights, a limiting unique solution is ultimately achieved, viz. the controlled-ﬂow state yielding maximal transport. For even larger sill heights no continuous ﬂow solutions exist. Boren¨ as and Lundberg [2] showed that by rescaling the variables (a ∗ ,b ∗ ) on the basis of r -1/2 , eqs. (1.1) and (1.2) could be perturbatively resolved using the expansion parameter r 1/2 . This yielded lowest-order results representing the classical solution for hydraulic ﬂow through a nonrotating channel. In the present study an alternative procedure based on ˆ D ∞ is introduced, resulting in a lowest-order problem corresponding to the upstream depth being sufﬁciently large to make the potential vorticity vanish, cf. Whitehead et al. [12]. ∗ Corresponding author, e-mail: janek@staff.ttu.ee c  2005 WILEY-VCHVerlag GmbH & Co. KGaA, Weinheim