86
T. Torsvik
where ρ 0 is a reference density, typically 1000 kg/m 3 for fresh water and around
1030 kg/m 3 for salt water, and ρ is the deviation from the reference level for a
particular volume of water. For internal waves on the interface between fresh and
salt water, which is the strongest density gradient likely to occur in the ocean, we
will have a density difference of ρ = 0.03ρ 0 , which gives us the maximum wave
speed of c 3D = 0.17c 2D for internal gravity waves. Normally such strong density
differences are not maintained for very long in the ocean, so for practical applications we may assume c 3D < 0.1c 2D .
From the numerical modelling point of view we see that the CFL condition imposed by surface gravity waves is stronger than the CFL condition associated with
internal gravity waves. Since the surface gravity waves can be calculated with the
use of 2D models, many ocean models apply a mode-split method where separate
model equations are used for the fast barotropic motion (2D) and the slow baroclinic motion (3D). This allows the model to use different time steps for the different modes, integrating the barotropic mode at a higher frequency than the baroclinic
mode but with a simpler set of model equations. This feature is built into many
ocean models, including the Rossby Center Ocean Model RCO that is thoroughly
discussed in Chap. 4 and the results from which are extensively used in Chaps. 9
and 10.
3.3.2 Model Equations on a Rotating Frame of Reference
The equations of motion in fluid dynamics express the rate of change of some quantity F (x, y, z, t) at each point in space following a fluid volume of fixed identity. The
Lagrangian description provides a natural frame of reference for such motions, but
the equations we wish to use are formulated within the Eulerian frame of reference.
For arbitrary increments dx, dy, dz, and dt the increment in F is
dF =
∂F
∂t
dt +
∂F
∂x
dx +
∂F
∂y
dy +
∂F
∂z
dz.
(3.19)
If, however, the increments are not arbitrary but are associated with the movement
of a small volume of fluid, the space and time increments are linked by the local
velocity components
dx = u(x, y, z, t)dt,
dy = v(x, y, z, t)dt,
dz = w(x, y, z, t)dt,
and we can write Eq. (3.19) as
dF
dt
=
∂F
∂t
+ u
∂F
∂x
+ v
∂F
∂y
+ w
∂F
∂z
.
(3.20)
In order to emphasize that the operation of time derivative is carried out while following a volume of fluid, the notation D/Dt is often used as replacement for d/dt
T. Torsvik
where ρ 0 is a reference density, typically 1000 kg/m 3 for fresh water and around
1030 kg/m 3 for salt water, and ρ is the deviation from the reference level for a
particular volume of water. For internal waves on the interface between fresh and
salt water, which is the strongest density gradient likely to occur in the ocean, we
will have a density difference of ρ = 0.03ρ 0 , which gives us the maximum wave
speed of c 3D = 0.17c 2D for internal gravity waves. Normally such strong density
differences are not maintained for very long in the ocean, so for practical applications we may assume c 3D < 0.1c 2D .
From the numerical modelling point of view we see that the CFL condition imposed by surface gravity waves is stronger than the CFL condition associated with
internal gravity waves. Since the surface gravity waves can be calculated with the
use of 2D models, many ocean models apply a mode-split method where separate
model equations are used for the fast barotropic motion (2D) and the slow baroclinic motion (3D). This allows the model to use different time steps for the different modes, integrating the barotropic mode at a higher frequency than the baroclinic
mode but with a simpler set of model equations. This feature is built into many
ocean models, including the Rossby Center Ocean Model RCO that is thoroughly
discussed in Chap. 4 and the results from which are extensively used in Chaps. 9
and 10.
3.3.2 Model Equations on a Rotating Frame of Reference
The equations of motion in fluid dynamics express the rate of change of some quantity F (x, y, z, t) at each point in space following a fluid volume of fixed identity. The
Lagrangian description provides a natural frame of reference for such motions, but
the equations we wish to use are formulated within the Eulerian frame of reference.
For arbitrary increments dx, dy, dz, and dt the increment in F is
dF =
∂F
∂t
dt +
∂F
∂x
dx +
∂F
∂y
dy +
∂F
∂z
dz.
(3.19)
If, however, the increments are not arbitrary but are associated with the movement
of a small volume of fluid, the space and time increments are linked by the local
velocity components
dx = u(x, y, z, t)dt,
dy = v(x, y, z, t)dt,
dz = w(x, y, z, t)dt,
and we can write Eq. (3.19) as
dF
dt
=
∂F
∂t
+ u
∂F
∂x
+ v
∂F
∂y
+ w
∂F
∂z
.
(3.20)
In order to emphasize that the operation of time derivative is carried out while following a volume of fluid, the notation D/Dt is often used as replacement for d/dt