We explore the nature of inertial equilibration of equatorial
flows in the presence of mean meridional and vertical shears of the
basic state, with oceanic applications in mind. The study is
motivated by the observational evidence that the subthermocline
equatorial mean circulation displays nearly zero Ertel potential
vorticity away from the equator, when taking into account the
non-traditional horizontal component of the Earth rotation. This
observed state precisely verifies the marginal condition for inertial
instability: a linear analysis for the equatorial β-plane
confirms that the usual condition of instability, namely that Ertel
potential vorticity should be of opposite sign to the vertical
Coriolis parameter, remains valid even when the traditional
approximation is relaxed. Analytical linear normal modes reveal that
a meridional shear of the basic state leads to a vertical stacking of
equatorially-trapped zonal flows of alternate signs, with a new
centre of symmetry located at the dynamical equator. A vertical shear
of the basic state causes a meridional stacking of extra-equatorial
zonal flows.
In an inviscid framework, a two-dimensional formulation is
ill-posed and we resort to non-hydrostatic viscous simulations to
determine the nonlinear normal forms of the system. The influence of
a small-scale eddy diffusivity and a large-scale Rayleigh damping on
the equilibrated vertical scale is determined numerically. The
nonlinear equilibration occurs through a steady-state bifurcation
from a basic state without jets to another steady state with
secondary jets of alternate signs. The final state corresponds to
eastward jets located on the geographic equator, while westward jets
are located near the dynamical equator. These results are consistent
with in situ observations of equatorial deep jets.
The analogy between the equatorial meridional shear flow and the
cylindrical Couette–Taylor flow with an axial density
stratification is detailed. There is a strong similarity in the
general symmetries and nonlinear normal forms of the two problems.
Similarly to the homogeneous Couette–Taylor flow, the gap width
between the two cylinders is important for determining the axial
scale of the secondary flow through the Reynolds number. For the
equatorial problem, an upper bound for the height scale of inertial
jets is such that the corresponding equatorial radius of deformation
times √2 fits between the geographic and dynamic equators.
One of our main conclusions is that the raison
d’être of the observed region of zero
Ertel potential vorticity is to facilitate angular momentum exchanges
between the two hemispheres and inertial deep jets are the byproducts
of this angular momentum mixing.