Introduction
95
advection must dominate vertical diffusion. Thus it must be advective,
nonlinear dynamics that primarily determines the vertical and horizontal
structure of the density field which in turn, through the effects of buoyancy and
rotation, shapes the motion field. The problem posed for explaining the density
and velocity structure with depth is fundamentally nonlinear, and the
theoretical challenge is keen.
The problem as it naturally presents itself to us involves the simultaneous
explanation of the vertical and horizontal structures of the density field and
renders the physics very difficult. It has been possible to effect great
simplifications in the theoretical treatment of the problem by employing
quasi-geostrophic models of the stratified, wind-driven ocean circulation.
These models essentially split the problem in two. Since the vertical structure of
the horizontal velocity field is related to the horizontal density gradient, we can
with these models concentrate first on describing the structure of the
horizontally varying part of the density field accepting, as given, the vertical
density gradient. This considerably simplifies the problem and, as we see below,
renders it nearly linear, although with important and essential nonlinear
aspects. This simplification comes naturally with quasi-geostrophic theory
(Pedlosky 1987) since in this theory the density field is linearized about a
known and horizontally and temporally constant vertical stratification. Thus
the density field is written:
Ptotal = Ps(z) + p(x, y, z, t)
(3.1.1)
where, p(x, y, z, t) is the density variation around the laterally averaged density
field, ps(z), which is a function only of depth. A principal feature and
requirement of quasi-geostrophic theory is that:
(3.1.2)
Actually, the total horizontal variation of the density over the scale of the
gyre is certainly as large as the vertical variation of the mean density, as can be
seen from Fig. 3.1.1. Therefore, the quasi-geostrophic model cannot be
expected to describe well the full density structure. Rather, since the velocity
depends only on the horizontal gradient of p, we can ask it only for a theory for
the vertical structure of the velocity field. A theory for the full vertical and
horizontal structure of the density field is termed thermocline theory, which we
take up in Chapter 4. We see below, however, that quasi-geostrophic theory
fares unexpectedly well in anticipating many of the results of the more complex
theory, especially for those aspects of the theory, such as the vertical variation
of the velocity field, that depend less on the overall variation of the density field
itself and pivot more on the local horizontal gradient of the density field.
Another simplification that has been of great value in illuminating the
basic physical ideas of the dynamics has been the use of layer models of the
ocean circulation. These models eliminate the continuous depth variable from
95
advection must dominate vertical diffusion. Thus it must be advective,
nonlinear dynamics that primarily determines the vertical and horizontal
structure of the density field which in turn, through the effects of buoyancy and
rotation, shapes the motion field. The problem posed for explaining the density
and velocity structure with depth is fundamentally nonlinear, and the
theoretical challenge is keen.
The problem as it naturally presents itself to us involves the simultaneous
explanation of the vertical and horizontal structures of the density field and
renders the physics very difficult. It has been possible to effect great
simplifications in the theoretical treatment of the problem by employing
quasi-geostrophic models of the stratified, wind-driven ocean circulation.
These models essentially split the problem in two. Since the vertical structure of
the horizontal velocity field is related to the horizontal density gradient, we can
with these models concentrate first on describing the structure of the
horizontally varying part of the density field accepting, as given, the vertical
density gradient. This considerably simplifies the problem and, as we see below,
renders it nearly linear, although with important and essential nonlinear
aspects. This simplification comes naturally with quasi-geostrophic theory
(Pedlosky 1987) since in this theory the density field is linearized about a
known and horizontally and temporally constant vertical stratification. Thus
the density field is written:
Ptotal = Ps(z) + p(x, y, z, t)
(3.1.1)
where, p(x, y, z, t) is the density variation around the laterally averaged density
field, ps(z), which is a function only of depth. A principal feature and
requirement of quasi-geostrophic theory is that:
(3.1.2)
Actually, the total horizontal variation of the density over the scale of the
gyre is certainly as large as the vertical variation of the mean density, as can be
seen from Fig. 3.1.1. Therefore, the quasi-geostrophic model cannot be
expected to describe well the full density structure. Rather, since the velocity
depends only on the horizontal gradient of p, we can ask it only for a theory for
the vertical structure of the velocity field. A theory for the full vertical and
horizontal structure of the density field is termed thermocline theory, which we
take up in Chapter 4. We see below, however, that quasi-geostrophic theory
fares unexpectedly well in anticipating many of the results of the more complex
theory, especially for those aspects of the theory, such as the vertical variation
of the velocity field, that depend less on the overall variation of the density field
itself and pivot more on the local horizontal gradient of the density field.
Another simplification that has been of great value in illuminating the
basic physical ideas of the dynamics has been the use of layer models of the
ocean circulation. These models eliminate the continuous depth variable from