6.12 Geostrophic Adjustment of a Density Front
151
∂h 1
∂t
= −
∂ (u 1 h 1 )
∂ x
(6.60)
where the Boussinesq approximation, ρ 2 /ρ 1 ≈ 1, has also been applied. These
equations are formally identical to those governing surface gravity waves (Eqs. 4.12
and 4.13). Under the assumption that interface displacements are small compared
with the thickness of the top layer, it can be shown that the wave solution of the
two-layer reduced-gravity model are interfacial waves of a phase speed of (e.g.,
Gill, 1982):
c iw =
g h 1,o
(6.61)
which is much smaller than the phase speed of long surface gravity waves. Note
that this is the same result that (4.27) gives for h 2 >> h 1 . Obviously, application of the reduced-gravity concept to the two-layer shallow-water wave equations
has filtere away fast propagating surface gravity waves. Since the phase speed of
internal waves is much smaller compared with that of surface gravity waves, the
reduced-gravity model allows for much longer time steps than the full two-layer
model, which makes this model attractive for studies of internal-wave propagation.
The CFL stability criterion for the resultant finite-di ference wave equations is:
Δt ≤ Δx/c iw
(6.62)
Derivation of the reduced-gravity equations for more than two layers remains for
the reader. FORTRAN codes of reduced-gravity layer models are not included in
this book.
6.12 Geostrophic Adjustment of a Density Front
6.12.1 Background
Density fronts are narrow zones in the ocean across which the density of seawater
changes rapidly. Almost uniform densities are found on either side of the frontal
zone. A variety of processes create density fronts, such a freshwater runoff from
rivers into the ocean. Once a density contrast is established, adjustment toward a
geostrophic steady-state circulation follows. To explore this geostrophic adjustment,
firs described by Rossby (1938), we consider the initial situation of a cylindrical
patch of low-density surface water surrounded by a denser ocean of uniform density
(Fig. 6.20).
151
∂h 1
∂t
= −
∂ (u 1 h 1 )
∂ x
(6.60)
where the Boussinesq approximation, ρ 2 /ρ 1 ≈ 1, has also been applied. These
equations are formally identical to those governing surface gravity waves (Eqs. 4.12
and 4.13). Under the assumption that interface displacements are small compared
with the thickness of the top layer, it can be shown that the wave solution of the
two-layer reduced-gravity model are interfacial waves of a phase speed of (e.g.,
Gill, 1982):
c iw =
g h 1,o
(6.61)
which is much smaller than the phase speed of long surface gravity waves. Note
that this is the same result that (4.27) gives for h 2 >> h 1 . Obviously, application of the reduced-gravity concept to the two-layer shallow-water wave equations
has filtere away fast propagating surface gravity waves. Since the phase speed of
internal waves is much smaller compared with that of surface gravity waves, the
reduced-gravity model allows for much longer time steps than the full two-layer
model, which makes this model attractive for studies of internal-wave propagation.
The CFL stability criterion for the resultant finite-di ference wave equations is:
Δt ≤ Δx/c iw
(6.62)
Derivation of the reduced-gravity equations for more than two layers remains for
the reader. FORTRAN codes of reduced-gravity layer models are not included in
this book.
6.12 Geostrophic Adjustment of a Density Front
6.12.1 Background
Density fronts are narrow zones in the ocean across which the density of seawater
changes rapidly. Almost uniform densities are found on either side of the frontal
zone. A variety of processes create density fronts, such a freshwater runoff from
rivers into the ocean. Once a density contrast is established, adjustment toward a
geostrophic steady-state circulation follows. To explore this geostrophic adjustment,
firs described by Rossby (1938), we consider the initial situation of a cylindrical
patch of low-density surface water surrounded by a denser ocean of uniform density
(Fig. 6.20).