Introduction
The analytical model
A numerical application
Summary
Mean mass transport induced by internal
Kelvin waves, with application to the circulation
in the Van Mijen fjord in Svalbard.
Eivind Støylen eivind.stoylen@geo.uio.no
& Jan Erik Weber j.e.weber@geo.uio.no
Department of Geosciences, MetOs section
University of Oslo
14.October 2008
Introduction
The analytical model
Outline
1
Introduction
Wave drift
Internal Kelvin waves
2
The analytical model
The geometry
Linear theory
Nonlinear theory
3
A numerical application
The Van Mijen fjord
4
Summary
Summary
Further research
A numerical application
Summary
Introduction
The analytical model
A numerical application
Summary
Wave drift
Stokes (1847):
A mean Stokes drift Ūs arise from
propagating wave motion
Movie: Wikipedia
Longuet-Higgins (1953):
The Lagrangian drift Ūl is the
sum of Ūs and a viscositydependent Eulerian mean Ūe .
Introduction
The analytical model
A numerical application
Summary
Internal Kelvin waves
Propagating on a
pycnocline along a
boundary, due to Coriolis
Amplitude decay
exponentially outwards to
the internal Rossby radius
a of order 1-10 km
Commonly observed in
large stratified lakes/fjords
Introduction
The analytical model
A numerical application
Summary
Previous work
Wind-induced internal Kelvin wave in a large stratified lake:
Csanady (1972):
Derives a linear large-amplitude internal wave, the first to
suggest significant mass transport near shore
Wunsch (1973):
From Csanady, derives Stokes drift in the linear wave.
Eulerian mean is neglected
Ou and Bennett (1979):
Derives the mean second order drift, retaining the viscous
Eulerian mean current. Comparison with observations in
Lake Kinneret, Israel
Gómez-giraldo et al. (2006):
Field data and numerical model in Lake Kinneret. Verifies
an internal Kelvin wave propagating around the basin
Introduction
The analytical model
A numerical application
Two-layer model
Hydrostatic pressure
distribution
H2 much larger than H1
u2 , v2 negligible
Ice lid;
η negligible
No-slip at z = H1
Variable under-ice
pressure Ps
Summary
Introduction
The analytical model
A numerical application
Summary
Tidally driven internal Kelvin wave
Monochromatic, constant
frequency ω
Coast at y = 0
Anticipate v1 = 0 everywhere
Guess ξ˜ = Ae−αx−y/a eiψ
Phase function: ψ = kx + ly − ωt
a: internal Rossby radius
α: horizontal damping
coefficient
Introduction
The analytical model
A numerical application
Summary
Linear theory
Separate u1 in barotropic and boundary-layer part u1 = ũ1 + û1 .
From first order Navier-Stokes and continuity:
ξ˜ = Ae−αx−y/a cos ψ
c1 A −αx−y/a ũ1 = −
e
cos ψ − sin ψ
H1
2
−(1−i)(H1 −z)/δ
û1 = <{−ũ1 e
}
Boundary-layer thickness δ =
Small parameter ≡ δ/(2H1 )
α is of order p
2ν/ω H1
Introduction
The analytical model
A numerical application
Summary
Nonlinear theory
Integrate;
Z
U=
Second order in wave steepness:
H1
u1 dz
ξ
Oscillatory and
mean part;
U = Ũ+ Ū, Ũ ∼ e
(x)
− τi )/ρ1
(y)
− τi )/ρ1
−f V̄ − c21 ξ¯x = HS1 + (τs
f Ū − c21 ξ¯y = HS2 + (τs
−iωt
Average over a wave
period 2π/ω
(x)
(y)
Ūx + V̄y = 0
HS1, HS2: Nonlinear wave forcing
terms
Introduction
The analytical model
A numerical application
Summary
Nonlinear theory
Integrate;
Z
U=
Second order in wave steepness:
H1
u1 dz
ξ
Oscillatory and
mean part;
U = Ũ+ Ū, Ũ ∼ e
(x)
− τi )/ρ1
(y)
− τi )/ρ1
−f V̄ − c21 ξ¯x = HS1 + (τs
f Ū − c21 ξ¯y = HS2 + (τs
−iωt
Average over a wave
period 2π/ω
(x)
(y)
Ūx + V̄y = 0
HS1, HS2: Nonlinear wave forcing
terms
Introduction
The analytical model
A numerical application
Summary
Solve mean second-order equations
From
Longuet-Higgins
(1953):
Ū = Ūl = Ūs + Ūe
Z
Ūs = H1 u1 dt · u1x
(V̄s = 0)
Friction as drag on
Eulerian current:
(x)
τs
= −cd |Ūe |Ūe /H12
Eulerian equations
−f V̄e − c21 ξ¯x = −
∂ 3
cd
c1 Ūs − 2 Ūe2
∂x 2
H1
f Ūe − c21 ξ¯y = 0
Ūex + V̄ey = −
∂
Ūs
∂x
Introduction
The analytical model
A numerical application
Summary
Solve mean second-order equations
From
Longuet-Higgins
(1953):
Ū = Ūl = Ūs + Ūe
Z
Ūs = H1 u1 dt · u1x
(V̄s = 0)
Friction as drag on
Eulerian current:
(x)
τs
= −cd |Ūe |Ūe /H12
Eulerian equations
−f V̄e − c21 ξ¯x = −
∂ 3
cd
c1 Ūs − 2 Ūe2
∂x 2
H1
f Ūe − c21 ξ¯y = 0
Ūex + V̄ey = −
∂
Ūs
∂x
Introduction
The analytical model
A numerical application
Solutions
c1 A2 −2(αx+y/a)
e
, V̄s = 0
2H
r 1
αH1
Ūe =
c1 Ae−αx−y/a
cd
Ūs =
At x = 0:
ũ1 = 0.18 m/s
ūl = 0.096 m/s
v̄l = O(10−3 ) m/s
r
A
αH1
−2y/a −2αx
−y/a −αx
V̄e = αac1 A
(1 − e
)e
+
(1 − e
)e
2H1
cd
r
A
αH1
ξ¯ =
Ae−2αx +
A(1 − e−y/a )e−αx
4H1
cd
Summary
Introduction
The analytical model
A numerical application
The Van Mijen fjord
Figure: Bergh (2004)
Summary
Introduction
The analytical model
A numerical application
Summary
Preliminary model result
The internal Kelvin Wave is
indeed present
Propagating along the
coast, damped by friction
Large amplitudes
at the sounds
Figure: Interface amplitudes
(Støylen 2008)
Parameter
T
∆ρ/ρ
∆x
η0
H1
Value
119 hrs
0.001
200 m
52.2 cm
30 m
Introduction
The analytical model
A numerical application
Summary
Drift pattern in the Van Mijen fjord?
Eastward flow at
southern coast
Friction ⇒ flow to
interior
Return flow at
northern wall
Mass conservation
Introduction
The analytical model
A numerical application
Summary
Internal Kelvin wave under ice in a two-layer system
Integrated between material boundaries to retain
Lagrangian properties
Alongshore: Eulerian flux comparable to Stokes flux
Normal to coast: Non-zero Eulerian flux due to friction
Model results suggest a tidally driven internal Kelvin wave
in the Van Mijen fjord
Summary
Introduction
The analytical model
A numerical application
Further research
Tidally driven model runs
Barotropic response, interaction between the sounds
Include wind stress in model
Set-up of internal waves? Upwelling scenarios
Include non-linear terms in the numerical model
Further developing the analytical model
Cooperation with Unis, Svalbard. Field measurements
Summary
Introduction
The analytical model
A numerical application
Summary
Thank you
Mean mass transport induced by internal
Kelvin waves, with application to the circulation
in the Van Mijen fjord in Svalbard.
Eivind Støylen eivind.stoylen@geo.uio.no
& Jan Erik Weber j.e.weber@geo.uio.no
Department of Geosciences, MetOs section
University of Oslo
14.October 2008
Additional references
Gómez-Giraldo, A., Imberger, J., Antenucci, J.P., 2006:
Spatial structure of the dominant basin-scale internal
waves in Lake Kinneret. Limnol. Oceanogr., 51, 229-246
Appendix: Parameters and values
Choice of values for plot:
A=10 m
Parameters:
c1 =
p
g0 H1
T=12.4 hrs
f = 1.42e-4 s−1
k = ω/c1
H1 =30m
a = c1 /f
∆ρ/ρ = 0.001
α = k/2
g = 9.81 m/s2
l = f /(2c1 )
ν = 3e-4 m2 /s
cd = 10−3
Appendix: Some assumptions and definitions
Navier-Stokes (N-S) in lower layer with large H2 and negligible
η:
0 = −g0 ξ˜x − Psx /ρ1 ,
g0 ≡ g(ρ2 − ρ1 )/ρ1
Surface pressure gradient balanced by interface gradient.
Upper layer N-S:
u1t = g0 ξ˜x + ν∇2 u1
fu1 = g0 ξ˜y
Geostrophy in y-direction. Assume frequency dependency:
˜ = −iω{ũ1 , û1 , ξ}
˜
∂/∂t{ũ1 , û1 , ξ}