Internal Hydraulic Jumps and Overturning Generated by Tidal Flow over a Tall
Steep Ridge
SONYA LEGG
Program in Atmosphere and Ocean Sciences, Princeton University, Princeton, New Jersey
JODY KLYMAK
School of Earth and Ocean Sciences, University of Victoria, Victoria, British Columbia, Canada
(Manuscript received 9 February 2007, in final form 6 February 2008)
ABSTRACT
Recent observations from the Hawaiian Ridge indicate episodes of overturning and strong dissipation
coupled with the tidal cycle near the top of the ridge. Simulations with realistic topography and stratification
suggest that this overturning has its origins in transient internal hydraulic jumps that occur below the shelf
break at maximum ebb tide, and then propagate up the slope as internal bores when the flow reverses. A
series of numerical simulations explores the parameter space of topographic slope, barotropic velocity,
stratification, and forcing frequency to identify the parameter regime in which these internal jumps are
possible. Theoretical analysis predicts that the tidally driven jumps may occur when the vertical tidal
excursion is large, which is shown to imply steep topographic slopes, such that dh/dxN/
 1. The vertical
length scale of the jumps is predicted to depend on the flow speed such that the jump Froude number is of
order unity. The numerical results agree with the theoretical predictions, with finite-amplitude internal
hydraulic jumps and overturning forming during strong offslope tidal flow over steep slopes. These results
suggest that internal hydraulic jumps may be an important mechanism for local tidally generated mixing at
tall steep topography.
1. Introduction
The flow of barotropic tides over ocean topography
may lead to the generation of internal tides and mixing,
a topic of much recent interest following observations
that about 30% of tidal dissipation occurs in the open
ocean (Egbert and Ray 2000). It is believed that some
of this energy loss results in mixing that could drive the
thermohaline circulation (Wunsch and Ferrari 2004).
Several recent field programs have examined the role
of tide–topography interactions in leading to this tidally
generated mixing at oceanic topography—which may
take the form of multiple irregular ridges, such as the
fracture zones around the Mid-Atlantic Ridge (Polzin
et al. 1997), or the form of isolated steep ridges, such as
the Hawaiian Ridge (Rudnick et al. 2003; Klymak et al.
2006)—and on continental slopes (Nash et al. 2007).
These observations have shown that tidal energy is both
converted into internal waves, which radiate away from
the topography, and used for local mixing at the topog-
raphy. To develop new energetically consistent param-
eterizations of tidal mixing for global climate models,
the physical processes governing the transfer of energy
from barotropic tide to baroclinic motion to mixing
need to be fully understood. This article describes one
physical process by which tidal flow over isolated to-
pography leads to local mixing, namely internal hydrau-
lic jumps, and explores the parameter regime in which
this behavior is possible.
As a prelude to this study, it is helpful to understand
the degree to which the parameter space of tidal flow
over isolated topography has been examined to date.
For a single ridge, the controlling dimensional param-
eters are
, the tidal frequency; N, the buoyancy fre-
quency; f, the Coriolis parameter; h0, the topographic
height; H, the water depth; L, a horizontal length scale
associated with the topography; and U0, the amplitude
of the barotropic current (Garrett and Kunze 2007;
Legg and Huijts 2006). From these a suitable choice of
Corresponding author address: Sonya Legg, Program in Atmo-
sphere and Ocean Sciences, Princeton University, 201 Forrestal
Road, Princeton, NJ 08544.
E-mail: sonya.legg@noaa.gov
SEPTEMBER 2008 L E G G A N D K L Y M A K 1949
DOI: 10.1175/2008JPO3777.1
© 2008 American Meteorological Society
JPO3777

governing nondimensional parameters consists of the
frequency ratios,
/f and
/N; the tidal excursion dis-
tance normalized by the topographic length scale, U0 /
(
L); relative height of the topography, h0 /H; and

 h0 Ls where s 


2
 f2
N2  2

12
, 1
so that  is the ratio of the topographic slope to the
slope of the internal tide characteristic. We will refer to
 as the topographic steepness, where supercritical
slopes have   1 and subcritical slopes have   1.
Another important parameter, although not indepen-
dent of those listed, is the topographic Froude number
Fr 
U0
Nh0
, 2
which is a measure of the obstruction of the flow by the
topography; in steady flows, strongly nonlinear lee
waves are generated for Fr  1. Previous studies have
examined the influence of increasing  and h0 /H on the
character of the internal tides and the energy conver-
sion (Balmforth et al. 2002; Llewellyn Smith and Young
2003; Khatiwala 2003; St. Laurent et al. 2003) and have
shown that energy conversion is increased by a factor of
2 as  increases from infinitesimally small to , and
increased still further as h0 /H → 1. Here U0 /(
L)  1
has been shown theoretically and numerically (Bell
1975; Legg and Huijts 2006) to lead to generation of
internal tides at higher harmonics of the forcing fre-
quency.
While theoretical studies have proved useful in pre-
dicting the energy conversion rate, they are unable to
examine regimes with finite-amplitude barotropic forc-
ing or the partitioning of energy between local mixing
and radiated internal tides. In Legg (2004) and Legg
and Huijts (2006), the parameter space of large U0 /
(
L) combined with small U0 /(Nh0) has been shown
numerically to lead to the possibility of nonlinear hy-
draulic effects, which may be one source of local mix-
ing. Another source of local mixing is the rapid dissi-
pation of internal tides of small vertical scale, which are
generated when   1, with a greater proportion of
energy in smaller wavelengths if h0 /H is small.
In this article, we further examine the possibility for
mixing associated with internal hydraulic jumps in a
somewhat different regime, where U0 /(
L) is small but
where strongly nonlinear transient lee waves with over-
turning similar to internal hydraulic jumps may form if
the Froude number associated with the vertical tidal
excursion distance is small.
The parameter space we are concerned with in this
paper is motivated by observations made close to to-
pography on Kaena Ridge during the Hawaiian Ocean
Mixing Experiment (Levine and Boyd 2006; Aucan et
al. 2006; Klymak et al. 2008). The top left of Figs. 1, 2
show two tidal cycles of observations made from R/P
FLIP near the top of the shelf break. While the total
water depth in this location is about 1000 m, R/P FLIP
observations only extended down to 800 m. Interesting
features of the observations include a sudden increase
in density around the time the current, dominated by
the M2 tide, changes sign, accompanied by almost ver-
tical isopycnals with overturning and high values of dis-
sipation. The enhanced dissipation and steepened
isopycnals extend well above the topography and are
clearly linked to the tidal cycle. Because the observa-
tions are made only in a single location, it is difficult to
identify the physical mechanism by which the tide leads
to this overturning and dissipation from these observa-
tions alone. We are therefore motivated to carry out a
numerical and theoretical study, whose purpose is to
identify these mechanisms, linking the processes taking
place at this location with those at other locations
around this topography and identifying the necessary
criteria for this phenomenon to exist, so that the un-
derstanding gained from the Hawaiian Ridge can be
generalized to other areas of the global ocean.
The Hawaiian Ridge falls into the region of param-
eter space characterized by   1, U0 /(
L) K 1, and
U0 /(Nh0) K 1, that is, region 5 in the parameter space
diagram of Garrett and Kunze (2007). The specific val-
ues of these parameters when the barotropic flow am-
plitude is around 5 cm s1 are 
4, U0 /(
L)
0.01
(corresponding to a horizontal tidal excursion of
around 350 m) and U0 /(Nh0)
0.006. Here, h0 /H is
large (i.e., h0 /H
0.8).
2. Model setup
In our model study we employ the Massachusetts
Institute of Technology general circulation model
(MITgcm; Marshall et al. 1997), integrating the nonhy-
drostatic Boussinesq equations in a 2D configuration
similar to that described in Legg and Huijts (2006) and
Khatiwala (2003). A nonhydrostatic model is prefer-
able because previous studies (e.g., Legg and Adcroft
2003) have shown that highly nonlinear internal bores
associated with rapid increases in density, which may be
responsible for the density perturbations seen in the
observations, are poorly represented by hydrostatic
models. Simulations in only two dimensions are a rea-
sonable first step for this topography because at the
Hawaiian Ridge the flow is constrained to go over the
topography through a channel between islands and can-
not go around the topography instead. The relative in-
1950 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y VOLUME 38