Formulation of the Homogeneous Model
27
circulation retains the character of the Sverdrup theory for the interior, and to
what degree the nature of the total circulation depends on the details of the
model of dissipation that we adopt.
This chapter considers the dynamics of the homogeneous model with this
aspect of the dynamics in mind as a preliminary to our study of stratified
models of the circulation. After a brief introduction and review of linear theory
we confront the issues associated with the conjunction of nonlinearity and
dissipation. The reader is referred to earlier reviews, such as Hendershott
(1987) and Pedlosky (1987), for a more complete discussion of the formulation
of the model and its relevant linear theory.
2.2 Formulation of the Homogeneous Model
There are actually at least two different ways to think about the homogeneous
model. We might think of a single layer of fluid in which the motion is exactly
two-dimensional and incompressible in which the rate of change of the total
vorticity, planetary plus relative vorticity, is produced by the wind-stress curl
acting as a body force in the layer, and in which the vorticity is dissipated on
each fluid element by dissipation mechanisms of various forms, for example,
proportional to the vorticity itself and/or due to the lateral diffusion of the
vorticity. The exact two-dimensionality of such a model allows the flow to be
written immediately in terms of a stream function 1/f.
We take another approach that leads to a governing equation with the
same structure but which is more consistent with the physical ideas discussed in
Chapter 1 and the models described in later chapters.
We consider the model shown in Fig. 2.2.1. A homogeneous layer of fluid
of constant thickness, D, lies beneath a mixed layer in which the turbulent
stresses excited by the wind stress drive an Ekman transport as described in
Chapter 1. Under the layer, and in contact with the bottom, lies another
viscous boundary layer in which viscous stresses produced by the motion
couple the fluid to the solid bottom.
We assume that the motion in all parts of the fluid is slow enough so that
the flow is in geostrophic balance everywhere. Further, we assume that the
north-south scale of the motion, though large enough to make the Rossby
number small, is at the same time small enough that the fJ plane approximation
is valid. This allows us to use Cartesian coordinates, but more importantly it
allows us to write the geostrophic balance for the momentum approximation in
terms of a characteristic value,.fo, of the Corio lis parameter in the gyre. Thus if:
fJL/fo « 1
(2.2.1)
the geostrophic relations for the eastward and northward velocities in the layer
of depth D can be written, to O(fJL/fo):
27
circulation retains the character of the Sverdrup theory for the interior, and to
what degree the nature of the total circulation depends on the details of the
model of dissipation that we adopt.
This chapter considers the dynamics of the homogeneous model with this
aspect of the dynamics in mind as a preliminary to our study of stratified
models of the circulation. After a brief introduction and review of linear theory
we confront the issues associated with the conjunction of nonlinearity and
dissipation. The reader is referred to earlier reviews, such as Hendershott
(1987) and Pedlosky (1987), for a more complete discussion of the formulation
of the model and its relevant linear theory.
2.2 Formulation of the Homogeneous Model
There are actually at least two different ways to think about the homogeneous
model. We might think of a single layer of fluid in which the motion is exactly
two-dimensional and incompressible in which the rate of change of the total
vorticity, planetary plus relative vorticity, is produced by the wind-stress curl
acting as a body force in the layer, and in which the vorticity is dissipated on
each fluid element by dissipation mechanisms of various forms, for example,
proportional to the vorticity itself and/or due to the lateral diffusion of the
vorticity. The exact two-dimensionality of such a model allows the flow to be
written immediately in terms of a stream function 1/f.
We take another approach that leads to a governing equation with the
same structure but which is more consistent with the physical ideas discussed in
Chapter 1 and the models described in later chapters.
We consider the model shown in Fig. 2.2.1. A homogeneous layer of fluid
of constant thickness, D, lies beneath a mixed layer in which the turbulent
stresses excited by the wind stress drive an Ekman transport as described in
Chapter 1. Under the layer, and in contact with the bottom, lies another
viscous boundary layer in which viscous stresses produced by the motion
couple the fluid to the solid bottom.
We assume that the motion in all parts of the fluid is slow enough so that
the flow is in geostrophic balance everywhere. Further, we assume that the
north-south scale of the motion, though large enough to make the Rossby
number small, is at the same time small enough that the fJ plane approximation
is valid. This allows us to use Cartesian coordinates, but more importantly it
allows us to write the geostrophic balance for the momentum approximation in
terms of a characteristic value,.fo, of the Corio lis parameter in the gyre. Thus if:
fJL/fo « 1
(2.2.1)
the geostrophic relations for the eastward and northward velocities in the layer
of depth D can be written, to O(fJL/fo):