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

Slope topography represents both a barrier to large-scale geophysical fluid transport as well as an important location of mesoscale feature generation. Standard quasi-geostrophic theory (Pedlosky, 1987) indicates that large-scale circulation features act in such a way as to conserve their potential vorticity, leading to the standard result of flow along (as opposed to across) topographic contours. Thus, slope topography creates a barrier between the open and coastal oceans, often inhibiting the transport of nutrient-rich waters into the coastal zone and at the same time trapping pollutants in the coastal zone.

As both numerical and observational approaches have limitations with respect to modeling the slope region, the objective of this brief communication is to provide an analytic framework for flow along slope topographies. Such a framework will serve as an idealized backbone upon which observational, numerical, experimental and further theoretical work can build and provide a point of comparison for better interpretation of the respective dynamics. In particular, the results presented have implications for cross-topography exchange and provide significant insight into the coupling between the slope bottom boundary layer and interior water column dynamics.

The problem formulation considered in this work follows that of Sansón
and van Heijst (2002), Kuehl (2014) and Kuehl and Sheremet (2014). A
rotating, single fluid layer is considered which flows along a sloping bottom
topography (i.e., along slope barotropic flow). The momentum equations and
continuity Eq. (

Kuehl (2014) provides a scaling analysis which justified Ekman dissipation
being the dominant dissipative term and relative vorticity being dominated by
cross-stream shear,

Kuehl (2014) considered the linear case of Eq. (

topography of the form

similarity variable

boundary conditions

initial condition

In nature, compressing (or intensifying) jets are often observed and an
analysis of ocean slope topography finds many locations where

For the expanding jet case, the analytical solution is valid over the domain

Motivated by the success and utility of the linear solutions provided above,
we seek a similarity solution for the nonlinear case (Eq.

Thus, for topography of the form

Equation (

It is possible to solve Eq. (

It can be seen that

Comparison between linear (open circles), nonlinear numerical (thick
dashed) and nonlinear analytic (solid lines) normalized transport functions.
Plotted is the ratio

The solutions presented above are relevant to barotropic, along slope flow
over generic topographies of the form

This is an analytical paper: the codes described are standard and easily reproduced from explanations provided in the text.

Preparation of this paper was led by RI and JK; however, the ideas contained herein are the result of numerous discussions between all the authors listed.

The authors declare that they have no conflict of interest.

This work was supported by the Texas General Land Office, Oil Spill Program (program manager: Steve Buschang) under TGLO contract no. 16-019-0009283 and the National Science Foundation, Physical Oceanography Program (program manager: Eric Itsweire) under grant no. 1823452. Edited by: Juan Restrepo Reviewed by: two anonymous referees