J. Fluid Mech. (2010), vol. 644, pp. 321–336. c© Cambridge University Press 2010
doi:10.1017/S0022112009992503
321
High-mode stationary waves in stratified flow
over large obstacles
JODY M. KLYMAK1†, SONYA M. LEGG2
AND ROBERT PINKEL3
1Department of Physics and Astronomy, University of Victoria, Victoria, Canada V8W 3P6
2Program in Atmosphere and Ocean Sciences, Princeton University, Princeton, NJ 08544, USA
3Scripps Institution of Oceanography, University of California, San Diego, La Jolla, CA 98105, USA
(Received 20 October 2008; revised 28 September 2009; accepted 28 September 2009)
Simulations of steady two-dimensional stratified flow over an isolated obstacle are
presented where the obstacle is tall enough so that the topographic Froude number,
Nhm/Uo  1. N is the buoyancy frequency, hm the height of the topography from
the channel floor and Uo the flow speed infinitely far from the obstacle. As for
moderate Nhm/Uo (∼1), a columnar response propagates far up- and downstream,
and an arrested lee wave forms at the topography. Upstream, most of the water
beneath the crest is blocked, while the moving layer above the crest has a mean
velocity Um =UoH/(H − hm). The vertical wavelength implied by this velocity scale,
λo = 2πUm/N , predicts dominant vertical scales in the flow. Upstream of the crest
there is an accelerated region of fluid approximately λo thick, above which there is
a weakly oscillatory flow. Downstream the accelerated region is thicker and has less
intense velocities. Similarly, the upstream lift of isopycnals is greatest in the first
wavelength near the crest, and weaker above and below. Form drag on the obstacle
is dominated by the blocked response, and not on the details of the lee wave, unlike
flows with moderate Nhm/Uo.
Directly downstream, the lee wave that forms has a vertical wavelength given by
λo, except for the deepest lobe which tends to be thicker. This wavelength is small
relative to the fluid depth and topographic height, and has a horizontal phase speed
cpx = −Um, corresponding to an arrested lee wave. When considering the spin-up to
steady state, the speed of vertical propagation scales with the vertical component of
group velocity cgz = αUm, where α is the aspect ratio of the topography. This implies
a time scale tˆ = tNα/2π for the growth of the lee waves, and that steady state is
attained more rapidly with steep topography than shallow, in contrast with linear
theory, which does not depend on the aspect ratio.
1. Introduction
Nonlinear lee waves are observed in the atmosphere as downslope flows over
mountain ranges (i.e. Lilly 1978) and in the ocean as flow over sills (i.e. Farmer &
Smith 1980). These flows have been studied in the field, laboratories, and numerically
(see Baines 1995 for a comprehensive review). These studies, however, have usually
been for flows where the lee wave is of similar scale to the obstacle (or even much
† Email address for correspondence: jklymak@uvic.ca

322 J. M. Klymak, S. M. Legg and R. Pinkel
500
–20 0 20 –20 0
0
Nhm/ Uo = 8.7
Nhm/ Um = 4.3
Nhm/ Uo = 86.7
Nhm/ Um = 43.3
2
U/Uo
Uo
Um
W
4
x(km)
20 –20 0
H
Dm
hm
20
(a) (b) (c)
1000
z(
m
)
1500
2000
Figure 1. (a) Steady state of moderate Nhm/Uo flow (i.e. low mode, but still blocked
upstream). (b) High Nhm/Uo flow (i.e. high mode) characteristic of tidal flow over deep
ocean ridges. (c) Sketch of flow parameters. The mean flow in all cases is in the positive-x
direction.
larger), meaning that in a finite-depth fluid they are dominated by low vertical modes
in the flow (i.e. figure 1a).
Recently, lee waves made up of high vertical modes have been observed at
underwater ridges and continental shelves. Large-amplitude nonlinear waves were
observed breaking near the crest of the underwater ridge between Oahu and Kauii
(Levine & Boyd 2006; Klymak, Pinkel & Rainville 2008). These breaking waves led to
vertical displacements of over 100 m, and drove large viscous dissipation and mixing.
Since the vertical wavelength was only a fraction of the local water depth and obstacle
height, we loosely term them ‘high mode’. Recent numerical work demonstrates that
these high-mode waves are formed at the topographic break during off-ridge tidal
flow (Legg & Klymak 2008). When the tide reverses the waves propagate on-ridge,
much like lee waves released from tidal flows over sills in fjords (Farmer & Smith
1980; Klymak & Gregg 2003). Similar phenomena have been implicated in mixing
on the Oregon continental slope (Nash et al. 2007).
In this paper, we step back from oscillating forcing and consider the response of a
large obstacle to steady forcing. The goal is to predict the lee response, in particular
the amplitude of the lee wave. We are also interested in understanding how long
it takes for the lee wave to establish itself, since, if the flow is oscillating and the
establishment time is large compared to the forcing period, the arrested lee wave may
not be observed.
We consider channel flow with a depth H , stratification N and an initial
barotropic velocity of Uo (figure 1c). For the flows investigated here, we use
oceanic values of N ≈ 5.2 × 10−3 s−1, H ≈ 2000 m and Uo ≈ 0.1 m s−1 for a very large
NH/Uo ≈ 100. The response of a stratified flow varies with the height of the obstacle
hm, as governed by the topographic Froude number Nhm/Uo. For a very small
obstacle, Nhm/Uo  1, linear dynamics predicts that stationary lee waves over the
obstacle will have the horizontal wavenumber kx set by the dominant topographic
wavenumber. The vertical wavenumber is set by the requirement that the waves be
arrested, so that the magnitude of the horizontal phase speed is equal to the flow
velocity: Uo = cpx = N/(k2x + k
2
z )
1/2. If the waves are hydrostatic this simplifies to kz =
N/Uo.