## Abstract

Intrinsic oscillations of stable geophysical surface frontal currents of the unsteady, nonlinear, reduced-gravity shallow-water equations on an *f* plane are investigated analytically and numerically. For frictional (Rayleigh) currents characterized by linear horizontal velocity components and parabolic cross sections, the primitive equations are reduced to a set of coupled nonlinear ordinary differential equations. In the inviscid case, two periodic analytical solutions of the nonlinear problem describing 1) the inertially reversing horizontal displacement of a surface frontal current having a fixed parabolic cross section and 2) the cross-front pulsation of a coastal current emerging from a motionless surface frontal layer are presented. In a linear and in a weakly nonlinear context, analytical expressions for field oscillations and their frequency shift relative to the inertial frequency are presented. For the fully nonlinear problem, solutions referring to a surface frontal coastal current are obtained analytically and numerically. These solutions show that the currents oscillate always superinertially, the frequency and the amplitude of their oscillations depending on the magnitude of the initial disturbance and on the squared current Rossby number. In a linear framework, it is shown that disturbances superimposed on the surface frontal current are standing waves within the bounded region, the frequencies of which are inertial/superinertial for the first mode/higher modes. In the same frame, a zeroth mode, which could be interpreted as the superposition of an inertial wave on a background vorticity field, would formally yield subinertial frequencies. For surface frontal currents affected by Rayleigh friction, it is shown that the magnitude of the mean current decays according to a power law and that the oscillations decay faster, because this decay follows an exponential law. Implications of the intrinsic oscillations and of their rapid dissipation for the near-inertial motion in an active ambient ocean are discussed.

*Corresponding author address:* Angelo Rubino, Institut für Meereskunde, Universität Hamburg, Troplowitzstr. 7, D-22529 Hamburg, Germany. Email: rubino@ifm.uni-hamburg.de