Articles | Volume 25, issue 1
Nonlin. Processes Geophys., 25, 201–205, 2018
Nonlin. Processes Geophys., 25, 201–205, 2018

Brief communication 06 Mar 2018

Brief communication | 06 Mar 2018

Brief communication: A nonlinear self-similar solution to barotropic flow over varying topography

Ruy Ibanez1, Joseph Kuehl2, Kalyan Shrestha3, and William Anderson3 Ruy Ibanez et al.
  • 1Mechanical Engineering Department, University of Rochester, Rochester, NY 14627, USA
  • 2Mechanical Engineering Department, University of Delaware, Newark, DE 19716, USA
  • 3Mechanical Engineering Department, University of Texas Dallas, Dallas, TX 75080, USA

Abstract. Beginning from the shallow water equations (SWEs), a nonlinear self-similar analytic solution is derived for barotropic flow over varying topography. We study conditions relevant to the ocean slope where the flow is dominated by Earth's rotation and topography. The solution is found to extend the topographic β-plume solution of Kuehl (2014) in two ways. (1) The solution is valid for intensifying jets. (2) The influence of nonlinear advection is included. The SWEs are scaled to the case of a topographically controlled jet, and then solved by introducing a similarity variable, η = cxnxyny. The nonlinear solution, valid for topographies h = h0 − αxy3, takes the form of the Lambert W-function for pseudo velocity. The linear solution, valid for topographies h = h0 − αxyγ, takes the form of the error function for transport. Kuehl's results considered the case −1 ≤ γ < 1 which admits expanding jets, while the new result considers the case γ < −1 which admits intensifying jets and a nonlinear case with γ = −3.

Short summary
We present a nonlinear analytic solution for barotropic flow relevant to the oceanographic slope region. A similarity approach is adopted and the solution takes the form of a Lambert W-function. A more general class of linear solutions is also discussed which take the form of error functions. The equations solved are similar to the heat equation and thus the results may be of interest beyond the geophysical community.