The two seminal studies on westward intensification, carried out by Stommel and Munk over 70 years ago, are revisited to elucidate the role of the domain aspect ratio (i.e., meridional to zonal extents of the basin) in determining the transport of the western boundary current (WBC). We examine the general mathematical properties of the two models by transforming them to differential problems that contain only two parameters – the domain aspect ratio and the non-dimensional damping (viscous) coefficient. Explicit analytical expressions are obtained from solutions of the non-dimensional vorticity equations and verified by long-term numerical simulations of the corresponding time-dependent equations. The analytical expressions as well as the simulations imply that in Stommel's model both the domain aspect ratio and the damping parameter contribute to the non-dimensional transport of the WBC. However, the transport increases as a cubic power of the aspect ratio and decreases linearly with the damping coefficient. On the other hand, in Munk's model the WBC's transport increases linearly with the domain aspect ratio, while the damping coefficient plays a minor role only. This finding is employed to explain the weak WBC in the South Pacific. The decrease in transport of the WBC for small-domain aspect ratio results from the decrease in Sverdrup transport in the basin's interior because the meridional shear of the zonal velocity cannot be neglected as an additional vorticity term.

As was noted by Henry Stommel, in the opening sentence of his
seminal 1948 study, “Perhaps the most striking feature of the general
oceanic wind-driven circulation is the intense crowding of
streamlines near the western borders of the oceans”. These strong
and narrow poleward-directed currents, often referred to as “western
boundary currents” (WBCs), counterbalance the weak and wide
equatorward (Sverdrup) flow in the interior of the basin. In the
North Atlantic this current is the Gulf Stream, and it was known to
oceanographers and explorers for a few centuries – see

Stommel, apparently in his first oceanography paper

In the last 70 years, both models have been modified and extended to
further explore the phenomenon of westward intensification in
different settings or to evaluate the importance of different
specific processes and terms in the governing equations

As in S48 and M50, a large number of these subsequent studies
employed the dimensional form of the governing equations which are
the time-independent rotating linearized shallow water equations
compounded by friction and forcing. These dimensional models include
numerous parameters: the zonal and meridional extents of the basin;
either the coefficient of linear drag (i.e., the coefficient in the
Rayleigh frictional term) or the kinematic eddy viscosity (i.e., the
coefficient in parameterization of the viscous term); the amplitude
(and possibly meridional structure) of the wind stress; the gradient
of Coriolis frequency (

By employing a non-dimensional approach,

The paper is organized as follows. Section

S48's dimensional vorticity equation for the spatial structure of
the stream function,

We begin by scaling Eq. (

From this point onwards, both the variables and the operators in the
differential equation(s) are non-dimensional, while dimensional
quantities will be accompanied by an asterisk (

As has been stated earlier, the non-dimensional formulation lumps
the five dimensional parameters in S48's model – zonal and
meridional extent of the basin, gradient of Coriolis frequency, wind
stress amplitude and Rayleigh friction coefficient – into just
two non-dimensional ones:

As is evident from Eq. (

The stream functions in different

In S48's model, we define the transport of the WBC as the product of
its width,

This expression will be compared below to its counterpart in M50's
model and will be compared in Sect.

Here, we note that the definition of the WBC's width is somewhat
arbitrary, and for definiteness we choose it to be

The non-dimensional counterpart of M50's vorticity equation,
obtained by employing the scaling proposed in this study in a
similar manner to that of S48 (refer to

Numerically obtained, non-dimensional stream functions for

Figure

We turn now to the estimation of the WBC's transport in M50's model.
As was done in S48's model, this transport is also defined as the
product of the boundary layer width (

As anticipated by

The numerical simulations described below were carried out using the
time-dependent, forced-dissipative, rotating shallow water equation
(SWE) dimensional solver that was successfully used in previous
studies. The solver employs the finite-difference method to solve
SWEs on the

The simulations presented here were carried out in a barotropic
ocean with the same characteristics as in S48, i.e., on an equatorial

Figure

Figure

Comparison between analytically (solid lines) and
numerically (dots) calculated values of transport (Tr) as a
function of

Figure

As is evident from Fig.

The non-dimensional transport of the western boundary
current (WBC) as a function of

Figure

Since the introduction of S48's and M50's models about 70 years
ago, numerous theoretical and numerical investigations have been
carried out to further explore the characteristics of westward
intensification

In this article, we address the issue raised by

In the traditional description of the S48 model the flow is
decomposed into two parts: a slow, anti-cyclonic flow in the
inner basin where the velocities are tiny so frictional effects can
be neglected, and a return boundary flow where the frictional
vorticity associated with the zonal shear of the poleward directed
velocity balances the planetary vorticity advected by this
velocity. According to this paradigm the WBC simply returns the
frictionless equatorward Sverdrup transport of the inner basin so
its transport is independent of the friction coefficient, and since
the (dissipation) Laplacian term does not affect the Sverdrup
interior flow, the transport of the WBC should also be independent
of the domain aspect ratio. The present study demonstrates that the
assumption of small damping,

To appreciate this subtle issue one should compare a square basin,
where

In M50's model the vorticity balance of the interior is more
involved since the bi-Laplacian dissipation operator (

The results derived here highlight an important effect that was overlooked in the classical (traditional) WBC theory, namely, the effect of the domain aspect ratio on the Sverdrup solution of the inner basin which results from the meridional shear of the zonal velocity in a narrow zonal channel.

The non-dimensional formulation presented here does not alter the
physical basis of the S48 and M50 models. We emphasize that the
dimensional transport (calculated from the product of the
non-dimensional transport and

The application of our results to the present-day ocean attributes the
small transport of the EAC compared to the other WBCs to the
geometry of the South Pacific ocean. In reality, factors other than
the domain aspect ratio may also be important in determining the
transport. For instance, the Brazil Current's volumetric transport
is low (especially in the northern part) because the current is
largely confined to the continental shelf

It is highly plausible that with a different arrangement of the continents in previous geologic times, the small domain aspect ratio that persisted in the ocean in those times was unable to support a strong WBC. Thus, the higher pole-to-Equator temperature gradient might have resulted from the smaller poleward heat flux by the WBC due to the decrease in the domain aspect ratio. This hypothesis should be addressed in a future work.

There are some typos in the expression for

For the reader's perusal, the variables in the aforementioned
equations are the same as the ones defined in

Corrections to

In the limit

To determine the zonal and meridional extents of a basin containing
a WBC, we identified the mean initiation and termination latitudes
of each WBC based on the available literature. The Gulf Stream
begins at the tip of Florida (

Dimensions of the gyres that contain the five western boundary currents in the present-day world ocean.

We define the meridional extent (

The mean dimensions

The numerical model used in this work can be downloaded from

No data were used or generated in this theoretical research.

KG performed the numerical simulation and the mathematical analyses, wrote the initial draft, and took part in writing all subsequent versions. NP conceived the basic theoretical idea, proofread all versions of the paper, and suggested the mathematical approach and numerical simulations. HG suggested the application to the five WBCs and proofread all versions of the paper.

The authors declare that they have no conflict of interest.

The authors thank David P. Marshall and another anonymous reviewer, whose comments were helpful in distilling some of the subtle points addressed in this work.

This research has been supported by the ISF-NSFC (grant no. 2547/17).

This paper was edited by Katsuro Katsumata and reviewed by David P. Marshall and one anonymous referee.