Searching for nonlinearity in internal tide wavebeams
N. Balmforth (Depts. Mathematics and Earth & Ocean Science, U.B.C.) and T. Peacock (Dept. Mechanical Engineering, M.I.T.)
(a) Source density
Background and mathematical model
5
Real
Imaginary
5
and
N2
ψxe−iωt
b = Re
iω
4
−2
1
2
3
2
4
6
8
10
(a)
1.5
1
2.5
12
0.5
X
X
2
3
−6
0
2.5
3.5
−4
1
−5
3.5
2
0
1.5
−0.5
1
ε=2.3
3
−1
0.5
ε=1.3
0
2.5
0
ε=3.35
↓
Linear, two-dimensional, inviscid gravity waves
(u, w) = Re (−ψz , ψx)e−iωt
0
2
2
The problem is formulated in terms of a streamfunction and buoyancy (density) perturbation,
2
3
1.5
2
4
6
X
8
10
12
1.2
(b)
1
0.8
0.6
0.4
0.2
0
−0.2
−1.5
−0.4
−2
−1.5
−1
−0.5
0
X
↑
1
↑
Conversion rate against ε
5
4
γ=2
γ=10
γ=4
γ=20
γ=1
3
γ=100
↓
ε=0.9
2
↑
↓
↑
↓
0
N2
∂ 2ψ ∂ 2ψ
−1
=
2
2
ω
∂z
∂x2
√
−1
N 2ω 2 − 1 to the horizontal; the phase and group velocThus waves travel at an angle tan
ities are orthogonal.
ε=1.75
−0.5
ε=3.9
ε=2.8
For an infinitely deep ocean, the waves generated at an obstable propagate off to z → ∞, and
we impose an outgoing radiation condition.
We distribute sources with density, Γ(x), along the topography and use a Green function,
G(x, x0, z, z 0), to write:
Z
ψ(x, z) = G(x, x0, z, h(x0))Γ(x0) dx0
(cf. Petrelis, Llewellyn Smith & Young, 2007, JPO, 36, 1053). The lower boundary condition
then furnishes an integral equation for Γ(x).
We use periodic topography in x: either a sinusoid or a “periodic Gaussian”,
sin(kx)
h(x) = h0 ×
exp[−γ(1 − cos kx)]
0.5
1
1.5
2
2.5
3
3.5
4
4.5
2
3
4
5
ε
6
7
8
9
10
5
Figure 1: Beam generation and interference for sinusoidal topography, and the (dimensionless)
energy conversion rate vs. , the ratio of topographic slope to wave slope.
Wavebeams leave the topography at the critical slopes; the beams scatter either up or
down and can interfere with one another constructively or destructively. This leads to
an energy conversion rate with a complicated dependence on the topographic height (or
relative maximum slope, ; cf. Nycander, 2006, JFM, 567, 415).
The experiment
The experiment consists of a tank of stratified salty fluid, with an obstable on the bottom that is dragged
back and forth sinusoidally with a given frequency and amplitude. The digital Schielern technique (Sutherland, Dalziel, Hughes & Linden, 1999, JFM, 390, 93) is used to image the internal wave field.
Figure 4: Comparison of theory and experiment.
We have, however, observed the generation of weak beams at twice the tidal frequency (the second harmonic).
Far-field representation
Above the crests of the topography, the Green function can be simplified and the solution
written in the form,
∞
X
Γp −ipµkz
e
+ constant,
ψ(x, z) =
p
p=1
p
where µ = N 2/ω 2 − 1 and the γp’s are certain integrals of Γ(x).
The wave structures are then no longer singular and can be compared with experiments.
1
Figure 2: Beam generation above periodic Gaussian ridges (and trenches), and
the conversion rate vs. for different bump separations, γ.
−1
where γ is a separation parameter (γ → 0 recovers the sinusoidal profile, and γ → ∞ gives an
isolated Gaussian).
Assuming the viscosity to be small, we can improve this solution to account for viscous smoothing of the wave beams using perturbation theory. This replaces the factor, e−ipµkz , with
"
#
νk 2N 4
exp −ipµk − 3 2
z
2
ω (N − ω )
0
ε=1.1
Boundary conditions and solution technique
ψ(x, h(x)) = U h(x)
γ=∞
1
0
0
On the topography:
1.5
6
↓
where ω is the tidal frequency and N the buoyancy frequency.
!
1
7
0.5
The internal wave equation is
0.5
ε=4.4
Conversion
The perturbations to the density gradient induced by these beams are formally singular in
inviscid theory, suggesting that they may immediately behave nonlinearly, suffering shear or
other kinds of instability, or creating density inversions that overturn the stratification.
4
Z
Linear inviscid theory of the tidally generated gravity waves indicates that these disturbance
can take the form of intense wave beams propagating away from the points on the topography
where the slope of the radiating gravity waves matches the slope of the bottom surface (e.g.
Balmforth, Ierley & Young, 2002, JPO, 32, 2900).
4
0
The wave beams
6
Z
Satellite altimetry has revealed that significant dissipation of the barotropic tide occurs in the
deep ocean in the vicinity of large topographic features (e.g. Egbert & Ray, 2000, Nature,
405, 775). This has been attributed to the generation of internal gravity waves as the tide
flows over such obstacles.
(b) bouyancy perturbation
We are still in search of more significant nonlinear effects...
Figure 3: A gaussian bump and the limiting knife edge.
The experiment runs at relatively low Reynolds number (of order 100). As a result, viscosity is crucial in
smoothing the wave beams and their amplitude becomes relatively weak (compare the viscous and inviscid
curves). Consequently, we have yet to observe any secondary instabilities or overturning in the wavebeams.