J. Math. Fluid Mech. 20 (2018), 255–261
2017 Springer International Publishing
Journal of Mathematical
Exact Steady Azimuthal Internal Waves in the f-Plane
Communicated by A. Constantin.
Abstract. We present an explicit exact solution of the nonlinear governing equations with Coriolis and centripetal terms in
the f-plane approximation for internal geophysical trapped waves with a uniform current near the equator. This solution
describes in the Lagrangian framework azimuthal equatorial internal waves propagating westward in a stratiﬁed rotational
Mathematics Subject Classiﬁcation. 76B55, 86A05, 76E30.
Keywords. Exact solution, Nonlinear geophysical wave, f-Plane.
We present an exact solution for internal geophysical trapped waves with Coriolis and centripetal forces
in the f-plane approximation in a stratiﬁed rotational ﬂuid. This three-dimensional nonlinear solution is
explicit in the Lagrangian framework. It prescribes the internal trapped waves with underlying a uniform
current and propagates westward within a restricted meridional range of approximately 2 degree latitude
from the equator. To study the problem of westward propagating geophysical trapped waves is interesting
in the equatorial region. It has been proposed that the interplay between equatorial wave and Equatorial
Undercurrent (EUC) in the ocean is one of the major generating mechanisms for El Nino and La Nina
Recently, some exact solutions describing nonlinear equatorial trapped waves in the Lagrangian frame-
work were obtained. These exact solutions are very important for understanding the nonlinear dynamics.
Constantin [2–5] presented explicit exact solution for three-dimensional equatorial internal waves in the
β-plane approximation. The solution of Constantin  can be reduced to Gerstner’s wave solution as it
ignores the Coriolis terms. Exact solution in Lagrangian variables was ﬁrst found for periodic travelling
waves over an inﬁnite depth in a homogenous ﬂuid by Gerstner , and rediscovered by Rankine .
The Gerstner’s solution could be modiﬁed to describe edge waves propagating over a sloping bed [1,29]
and was also adapted to describe the geophysical edge waves [17,24], building on the approach in Con-
stantin . Subsequent to , a vast variety of Gerstner-type exact, explicit and nonlinear solutions, which
model a number of interesting studies within the class of geophysical waves, were derived and analyzed in
[10,12,13,15,16,18,21]. Constantin and Johnson  presented some exact solutions, representing purely
azimuthal ﬂows, in the spherical and cylindrical coordinates. Hsu and Martin  prove the existence of
free-surface capillary-gravity azimuthal equatorial ﬂow in the cylindrical coordinates which allow for the
variation in the vertical direction but with no meridional ﬂow.
The presence of strong currents in the Equatorial Paciﬁc is well-documented; cf. Philander . The
exact solution for equatorial trapped waves with an underlying current was derived by Henry , Hsu
[18,19] and Kluczek . Henry and Sastre-Gomes  and Sanjurjo and Kluczek  discussed the
mean ﬂow velocities and mass transport induced by certain equatorial water waves. Sanjurjo present
rigorous mathematical results for equatorial waves with current. Recently, Henry  present an exact and
explicit nonlinear solution of a β-plane approximation for the equatorial trapped waves to the governing