3.17 Scaling
61
3.16.2 Boundary Conditions for Oceanic Applications
Wind stress operates as a frictional force at the sea surface. The associated boundary
condition is given by:
A z
∂u
∂z
z=0
=
τ
wind
x
ρ o
and
A z
∂v
∂z
z=0
=
τ
wind
y
ρ o
(3.66)
where ρ o is surface density. The components of the wind-stress vector are given by:
τ
wind
x
= ρ air C d U
U 2 + V 2 and τ
wind
y
= ρ air C d V
U 2 + V 2
(3.67)
where ρ air is air density, C d is the nondimensional wind-drag coefficien with values
of the order of 1.1–1.5×10
−3
, and U and V are horizontal components of the wind
vector measured at a height of 10 m above sea level.
Bottom friction is usually treated by either a linear or a quadratic approach. The
linear approach reads:
τ
bot
x
ρ o
=
A z
∂u
∂z
z=−H
= r lin u and
τ
bot
y
ρ o
=
A z
∂v
∂z
z=−H
= r lin v
(3.68)
where H is total depth of the water column, the friction parameter r lin has units
of metres per second, and (u, v) is the lateral f ow in vicinity of the seafloo . The
quadratic bottom-friction law is given by:
τ
bot
x
ρ o
= r u
u 2 + v 2 and
τ
bot
y
ρ o
= r v
u 2 + v 2
(3.69)
where r is a nondimensional bottom-drag coefficien . Other boundary conditions
include the vertically integrated form of the continuity equation (3.11) that we need
to predict sea-level elevation and associated barotropic pressure gradients. Source
and sink boundary terms associated with volume changes such as precipitation need
to be added to this equation, if required. Also required are boundary conditions
describing density changes owing to surface density flu es.
3.17 Scaling
3.17.1 The Idea
The idea behind the scaling theory is that we can estimate the relative significanc
of terms in the Navier–Stokes equations by using typical magnitudes or scales of
fl w variables. For instance, take a surface wave in the ocean that has a certain
period T and wavelength λ. If we want to model this wave, an obvious question
61
3.16.2 Boundary Conditions for Oceanic Applications
Wind stress operates as a frictional force at the sea surface. The associated boundary
condition is given by:
A z
∂u
∂z
z=0
=
τ
wind
x
ρ o
and
A z
∂v
∂z
z=0
=
τ
wind
y
ρ o
(3.66)
where ρ o is surface density. The components of the wind-stress vector are given by:
τ
wind
x
= ρ air C d U
U 2 + V 2 and τ
wind
y
= ρ air C d V
U 2 + V 2
(3.67)
where ρ air is air density, C d is the nondimensional wind-drag coefficien with values
of the order of 1.1–1.5×10
−3
, and U and V are horizontal components of the wind
vector measured at a height of 10 m above sea level.
Bottom friction is usually treated by either a linear or a quadratic approach. The
linear approach reads:
τ
bot
x
ρ o
=
A z
∂u
∂z
z=−H
= r lin u and
τ
bot
y
ρ o
=
A z
∂v
∂z
z=−H
= r lin v
(3.68)
where H is total depth of the water column, the friction parameter r lin has units
of metres per second, and (u, v) is the lateral f ow in vicinity of the seafloo . The
quadratic bottom-friction law is given by:
τ
bot
x
ρ o
= r u
u 2 + v 2 and
τ
bot
y
ρ o
= r v
u 2 + v 2
(3.69)
where r is a nondimensional bottom-drag coefficien . Other boundary conditions
include the vertically integrated form of the continuity equation (3.11) that we need
to predict sea-level elevation and associated barotropic pressure gradients. Source
and sink boundary terms associated with volume changes such as precipitation need
to be added to this equation, if required. Also required are boundary conditions
describing density changes owing to surface density flu es.
3.17 Scaling
3.17.1 The Idea
The idea behind the scaling theory is that we can estimate the relative significanc
of terms in the Navier–Stokes equations by using typical magnitudes or scales of
fl w variables. For instance, take a surface wave in the ocean that has a certain
period T and wavelength λ. If we want to model this wave, an obvious question