52
DYNAMICAL OCEANOGRAPHY
In this chapter the basis for a quantitative description of ocean currents
will be provided. After an introduction into the relevant temporal and
spatial scales in section 3.1, the governing equations of ocean flows, representing the momentum, mass, heat and salt balances are given in section
3.2. These are the equations of ‘classical physics’, describing the flow of a
rotating stratified layer of ocean water on a sphere. It comes as no surprise
that the general equations are much too complicated to solve analytically
and even with numerical solution techniques all the relevant scales of motion cannot be resolved on current supercomputers. Fortunately there are
many flow phenomena in the ocean that can be described by less complicated balances in the flow field. These phenomena can be understood
using simplified forms of the equations. As explained in section 3.3, with
help of the specific time scales of processes one can a priori identify some
dominant balances.
3.1. A priori scales
From the description of the ocean basins and the surface forcing in the previous
chapters we know that the ocean flows occur in a relatively thin layer of rotating
liquid, that the liquid is stratified, that the flows are forced at the surface by wind
stress and buoyancy flux, and that the domain is bounded by continental geometry
and bottom topography.
It is important to consider characteristic time scales of each of the processes
involved in the circulation. As we will see later, processes with comparable characteristic time scales are able to balance. The characteristic time scale of a particular process is also directly related to the contribution of that process to the
vorticity balance of the flow. Note that if the velocity vector of the flow is indicated by v, the vorticity vector is given by ω = ∇∧v (where ∧ indicates the
vector cross product) and has therefore a dimension of s −1 . A process with a relatively small characteristic time scale hence provides a larger contribution to the
vorticity balance in the flow than a process with a larger characteristic time scale.
3.1.1. Geometry
Let L and U be characteristic horizontal length and velocity scales of a particular ocean flow. For example, L could be the width of an ocean basin (L =
O(10 6 ) m) and U a maximum depth-averaged horizontal velocity in the basin (U
= O(10 −2 ms −1 )). The advective time scale τ a associated with the flow is then
given by
τ a =
L
U
.
(3.1)
DYNAMICAL OCEANOGRAPHY
In this chapter the basis for a quantitative description of ocean currents
will be provided. After an introduction into the relevant temporal and
spatial scales in section 3.1, the governing equations of ocean flows, representing the momentum, mass, heat and salt balances are given in section
3.2. These are the equations of ‘classical physics’, describing the flow of a
rotating stratified layer of ocean water on a sphere. It comes as no surprise
that the general equations are much too complicated to solve analytically
and even with numerical solution techniques all the relevant scales of motion cannot be resolved on current supercomputers. Fortunately there are
many flow phenomena in the ocean that can be described by less complicated balances in the flow field. These phenomena can be understood
using simplified forms of the equations. As explained in section 3.3, with
help of the specific time scales of processes one can a priori identify some
dominant balances.
3.1. A priori scales
From the description of the ocean basins and the surface forcing in the previous
chapters we know that the ocean flows occur in a relatively thin layer of rotating
liquid, that the liquid is stratified, that the flows are forced at the surface by wind
stress and buoyancy flux, and that the domain is bounded by continental geometry
and bottom topography.
It is important to consider characteristic time scales of each of the processes
involved in the circulation. As we will see later, processes with comparable characteristic time scales are able to balance. The characteristic time scale of a particular process is also directly related to the contribution of that process to the
vorticity balance of the flow. Note that if the velocity vector of the flow is indicated by v, the vorticity vector is given by ω = ∇∧v (where ∧ indicates the
vector cross product) and has therefore a dimension of s −1 . A process with a relatively small characteristic time scale hence provides a larger contribution to the
vorticity balance in the flow than a process with a larger characteristic time scale.
3.1.1. Geometry
Let L and U be characteristic horizontal length and velocity scales of a particular ocean flow. For example, L could be the width of an ocean basin (L =
O(10 6 ) m) and U a maximum depth-averaged horizontal velocity in the basin (U
= O(10 −2 ms −1 )). The advective time scale τ a associated with the flow is then
given by
τ a =
L
U
.
(3.1)