Geoscience Reference
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u
i
=
motion remains laminar and the turbulent viscosity
. If the damping compo-
nent
u
Ti
overcomes a certain limit, Re
wave
exceeds the critical value and the wave motion
becomes turbulent as known from experiments (
Babanin
,
2006
;
Babanin & Haus
,
2009
;
Dai
et al.
,
2010
). Then, turbulent stresses
ν
t
=
0
(
0
)
u
w
=
ν
t
du
orb
dz
τ
=
(9.30)
will significantly and rapidly enhance the effective total viscosity which is now a combi-
nation of the eddy and molecular viscosity:
ν
tot
=
ν
t
+
ν.
(9.31)
ν
t
produced by the resultant mean
wave velocity
u
i
in
(9.29)
. Note that this mean velocity is different from the ideal mean
orbital velocity
u
orb
,
i
as it takes into account the fact that the turbulent diffusion feeds back
to the wave motion and therefore contributes to a wave-energy damping.
As noticed above, the wave motion can be sufficiently represented by the linear-theory
outcomes. Keeping in mind that the existence of
u
Ti
is physically and mathematically
responsible for the appearance of turbulence in the first place (in a non-viscous fl
uid
u
T
i
=
0)
, it
was f
urther assumed that the magnitude of
u
Ti
is negligible compared to
u
orb
,
i
, i.e.
u
i
Now, it is possible to define the turbulent viscosity
≈
u
orb
,
i
. This way, it is possible to obtain
ν
t
by using the shear frequency
M
du
x
dz
2
du
z
dx
2
M
2
M
x
,
z
=
=
+
,
(9.32)
based on the idealised motion
u
orb
,
i
, which will be subsequently returned to the mean-flow
model with a time scale coarser than
dt
. Using linear wave theory (e.g.
Young
,
1999
), the
mean square value (integrated over wavelength) of the shear frequency for the wave motion
can be expressed as
π
2
kH
s
T
sinh
(
k
(
z
+
d
))
M
wave
(
z
)
=
.
(9.33)
sinh
(
kd
)
Circulation models simulate the mean flow, and turbulence models interconnect these
mean currents
U
and their fluctuations
U
, through the turbulent kinetic energy equation
for the evolution of TKE (denoted as
K
here, see
Burchard
et al.
,
1999
):
∂
K
U
i
∂
K
T
mean
+
X
i
=
D
K
+
P
S
+
G
−
E
K
(9.34)
∂
∂
where
U
i
is the mean-current component for
i
coordinate,
G
is the production of TKE
by buoyancy,
D
K
is the turbulent and viscous transport term and
E
K
is the dissipation
term. The temporal and spatial resolution
dT
mean
and
X
i
(capital letters), correspondingly,
indicate an equation for the mean flow (which simulates the process on scales that are
coarser than the scales
dt
,
dx
i
needed to simulate an individual wave).
P
S
, in the default
formulation of TKE
equation (9.34)
, signifies TKE production by mean currents:
P
current
=
ν
t
M
current
,
P
S
=
(9.35)
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