Les courants moyens – quel rôle jouent

Transcription

Netherlands Institute for Sea Research, Texel, Pays-Bas
Les courants moyens – quel rôle jouent-ils
dans l’effet Doppler ?
Theo Gerkema, Leo Maas & Hans van Haren
24 mai 2013
OGOA Lyon 2013
1/13
Les courants moyens – quel rôle jouent-ils dans l’effet Doppler ?
Internal waves and mean flows
Common situation:
internal waves in presence of (meso-scale) background flows.
How does that flow affect the wave’s frequency?
In various forms, one finds the following assertion in the literature:
• "Advection leads to a Doppler shift." (Kunze, 1985)
• "A background flow (. . . ) provides a way to increase the apparent
frequency of near-inertial waves through Doppler shifting."
(Zhai et al., 2005)
• ". . . a Doppler shift in the frequency of the inertial wave brought about
by the mean flow past the mooring." (White, 1972)
• . . . observations of horizontal velocity are computed from data recorded
on instrumented moored buoys. The mean drift causes non negligible
Doppler shifts in the observed frequencies (given by kU)."
(Frankignoul, 1970)
• "a constant current (. . . ) implies a Doppler shift." (Olbers, 1981)
OGOA Lyon 2013
2/13
Where does this idea come from?
With background flow U, the first terms of the linearized
equations of motion read
∂
∂
+U +··· = ···
∂t
∂x
Substituting a wave solution sin(ωt − kx), with wavenumber k and
wave frequency ω, we find the combination of terms
ω − Uk + · · · = · · ·
One can then define ω 0 = ω − Uk, the intrinsic frequency.
In the oceanographic literature, it is common to refer to Uk as a
"Doppler shift" (Bretherton & Garrett 1968, Kunze 1985, etc.)
OGOA Lyon 2013
3/13
"Doppler shifts" – a confusion of tongues?
"Doppler shifts" – a confusion of tongues?
In physics, "Doppler shifts" refer to
a difference in frequency between source and observer;
here mean flows never create a "Doppler shift".
In oceanographic parlance, "Doppler shifts" Uk refer to
a difference in frequency between two observers;
here mean flows always (by definition!) create a "Doppler shift".
. . . and sometimes the two are mixed up. . .
In the oceanographic context, one often compares frequencies of
sources (e.g. tidal) with observed ones (e.g. in moorings)
– one should then not apply the expression Uk !
To avoid confusion I shall call the former "Doppler shift";
the latter, "quasi-Doppler shift".
OGOA Lyon 2013
4/13
Doppler shifts revisited
Doppler (1842, 1846):
how do movements of source and observer affect the frequency?
He found that the frequency of
the source, ωs , is shifted to
ω = ωs
c − vo
c − vs
c wave speed
vs velocity of source
vo velocity of observer
There is a Doppler shift only if
source and observer move
relative to each other (vs 6= vo )
OGOA Lyon 2013
5/13
The theory was put to the test by Buys
Ballot in 1845. He had musicians play
trumpets on a steam train. Result:
change in tone agrees with theory.
Doppler (1847):
the effect of a steady uniform flow
Source and observer both at rest.
Wavelength changes (see figure).
Successive wave crests propagate
under identical circumstances and
thus need equal times to bridge
distance from source to observer.
Hence observed frequency equal to
frequency of emittance
→ no Doppler shift.
All cases collected in general formula by Bateman (1917):
ω = ωs
c + U − vo
c + U − vs
So if vs = vo , we have no Doppler shift – irrespective of mean flow U.
OGOA Lyon 2013
6/13
Analytical example
Analytical example
Linear long interfacial waves in presence of mean flow U.
NB: no wave dispersion included at this point!
Governing equations:
∂u
∂u
∂η
+U
= − g0
∂t
∂x
∂x
∂η
∂η h1 h2 ∂u
+U
+
= F(t, x)
∂t
∂x
h ∂x
with shear u, reduced gravity g0
and total water depth h:
g0 = g
ρ2 − ρ1
;
ρ∗
h = h1 + h2 .
A moving source explicitly included: forcing term F.
OGOA Lyon 2013
7/13
Analytical example
Specify source: oscillating at frequency ωs and moving at speed vs :
F(t, x) = sin(ωs t) G0 (x − vs t)
for an arbitrary function G (expressing F as its derivative).
Transform equations to frame of reference moving with the flow, and
combine the resulting equations:
2
∂F
∂2 η
2∂ η
−
c
=
,
∂t2
∂x2
∂t
with c = (g0 h1 h2 /h)1/2 , the phase speed of proper wave propagation.
For a system starting from rest, the solution reads
1
η=
2c
OGOA Lyon 2013
Z t
dτ
0
8/13
Z x+c(t− τ )
x−c(t− τ )
dξ
∂F
(τ, ξ ) .
∂τ
Analytical example
The inner integral can be solved, and after a transformation back to
the original system, we obtain
1
η (t, x) =
2
Z t
0
n
o
dτ sin(ωs τ ) G0 (X− ) + G0 (X+ ) ,
with X± = x − (±c + U )t + (±c + U − vs )τ.
The remaining integral can be solved analytically if we choose a
(moving) point source, described by a delta-distribution: G0 = δ.
The end result is:
η = η− + η+
with
η± =
h (±c + U )t − x i
1
sin ωs
2 | ± c + U − vs |
±c + U − vs
for all x satisfying (±c + U )t < x < vs t or vs t < x < (±c + U )t;
for all other x, η± = 0.
OGOA Lyon 2013
9/13
Analytical example
Setting:
two fixed observers, left and
right to initial position of
source:
Solution for several cases with
mean flow or/and moving
source:
→ mean flow creates
no Doppler shift
OGOA Lyon 2013
10/13
Analytical example
The solution can be easily modified to include a movement of the
observer, at velocity vo , via x = xo + vo t. Then:
ω=
± c + U − vo
ωs ,
± c + U − vs
k=
ωs
.
±c + U − vs
Returning now to quasi-Doppler shifts:
From the above, it follows that a fixed observer (vo = 0) measures
±c + U
ω=
ωs
± c + U − vs
and an observer moving with the mean flow (vo = U):
±c
ω0 =
ωs
±c + U − vs
Combining these with the expression for k:
ω 0 = ω − Uk
(quasi-Doppler shift)
Expression is invariant for movement of source (vs ), thus hiding
the fact that ω 0 , ω and k – each individually – depend on it !
OGOA Lyon 2013
11/13
Dispersion
Dispersion
So far wave dispersion was not considered.
How does this change the Doppler relations?
Example: source moving toward an observer at rest.
One can still argue that
C
ω=
ωs
C − vs
but the phase speed C is no longer known a priori !
C depends on ω → dispersion relation needed to close the problem.
E.g., for ω = Uk + (f 2 + c2 k2 )1/2 , i.e. Coriolis disperion & mean flow:
C± =
−(vs f 2 − Uωs2 ) ± ωs [c2 (ωs2 − f 2 ) + (U − vs )2 f 2 ]1/2
ωs2 − f 2
featuring movement of source vs and mean flow U.
OGOA Lyon 2013
12/13
Conclusions
Conclusions
1
It is important to distinguish Doppler shifts
– difference in frequency between source and observer –
from quasi-Doppler shifts ("Uk")
– difference in frequency between two observers.
2
A Doppler shift in frequency occurs only if source and observer
move relative to each other.
3
The presence of a steady mean flow does not change this fact !
4
This remains so if wave dispersion is included.
5
A mean flow does affect wavelength and amplitude!
Reference:
Gerkema et al. (2013), J. Phys. Oceanogr. 43, 432-441.
OGOA Lyon 2013
13/13

Les courants moyens – quel rôle jouent

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