58
DYNAMICAL OCEANOGRAPHY
However, many aspects of the ocean circulation theory developed in later chapters
are eventually independent of the representation of mixing processes and hence
using the simplest one is useful. It may therefore not come as a surprise that
values of the mixing coefficients are not well-known. A rough estimate of the
momentum mixing coefficient is given by the product of the relevant length scale
and velocity scale in the flow. However, in realistic flows these are usually hard to
determine. Because horizontal and vertical length scales differ considerably in the
ocean and the ocean is strongly stratified (inhibiting vertical mixing) horizontal
mixing coefficients are usually orders of magnitude larger than vertical ones. We
use the subscripts H and V to indicate the different coefficients, i.e., A V and A H
for the vertical and horizontal mixing coefficients of momentum.
In this first-order closure theory, the mixing of momentum is represented by
F I = ∇·T,
(3.17)
with T the part of the stress tensor representing shear. The general form of T is
T = A H (∇ H ⊗ v +(∇ H ⊗ v)
T )+A V (∇ z ⊗ v +(∇ z ⊗ v)
T ),
(3.18)
where the superscript T indicates the transpose, ∇ H is the horizontal gradient
operator and ∇ z =(0, 0,∂/∂z). The notation ⊗ is the dyadic product
a ⊗ b =
⎛
⎝
a 1 b 1 a 1 b 2 a 1 b 3
a 2 b 1 a 2 b 2 a 2 b 3
a 3 b 1 a 3 b 2 a 3 b 3
⎞
⎠ .
(3.19)
Estimates of A H are within the range 10 − 10 5 m 2 s −1 and A V varies from
values of 10 −5 m 2 s −1 in the deep ocean to values of 10 −1 m 2 s −1 in the upper
layer. In the same way the mixing of heat and salt is represented as
F T = ρ 0 C p (∇ H · (K H ∇ H T )+
∂
∂z
(K V
∂T
∂z
)),
(3.20a)
F S = ρ 0 (∇ H · (K H ∇ H S)+
∂
∂z
(K V
∂S
∂z
)),
(3.20b)
with estimates K V =10 −5 – 10 −4 m 2 s −1 and K H =10– 10 3 m 2 s −1 .
Some form of friction is needed to be able to satisfy boundary conditions, for
example no-slip (zero tangential and normal velocity) conditions on the continental boundaries and on the bottom topography. In many cases, this leads to
boundary layers (e.g., the Ekman layers) whose thickness depends on the mixing
coefficients. Friction introduces characteristic time scales
τ
H
w =
L 2
A H
; τ
V
w =
D 2
A V
.
(3.21)
In vorticity terms: random transport of momentum leads to diffusion of vorticity,
with a magnitude proportional to the mixing coefficients.
DYNAMICAL OCEANOGRAPHY
However, many aspects of the ocean circulation theory developed in later chapters
are eventually independent of the representation of mixing processes and hence
using the simplest one is useful. It may therefore not come as a surprise that
values of the mixing coefficients are not well-known. A rough estimate of the
momentum mixing coefficient is given by the product of the relevant length scale
and velocity scale in the flow. However, in realistic flows these are usually hard to
determine. Because horizontal and vertical length scales differ considerably in the
ocean and the ocean is strongly stratified (inhibiting vertical mixing) horizontal
mixing coefficients are usually orders of magnitude larger than vertical ones. We
use the subscripts H and V to indicate the different coefficients, i.e., A V and A H
for the vertical and horizontal mixing coefficients of momentum.
In this first-order closure theory, the mixing of momentum is represented by
F I = ∇·T,
(3.17)
with T the part of the stress tensor representing shear. The general form of T is
T = A H (∇ H ⊗ v +(∇ H ⊗ v)
T )+A V (∇ z ⊗ v +(∇ z ⊗ v)
T ),
(3.18)
where the superscript T indicates the transpose, ∇ H is the horizontal gradient
operator and ∇ z =(0, 0,∂/∂z). The notation ⊗ is the dyadic product
a ⊗ b =
⎛
⎝
a 1 b 1 a 1 b 2 a 1 b 3
a 2 b 1 a 2 b 2 a 2 b 3
a 3 b 1 a 3 b 2 a 3 b 3
⎞
⎠ .
(3.19)
Estimates of A H are within the range 10 − 10 5 m 2 s −1 and A V varies from
values of 10 −5 m 2 s −1 in the deep ocean to values of 10 −1 m 2 s −1 in the upper
layer. In the same way the mixing of heat and salt is represented as
F T = ρ 0 C p (∇ H · (K H ∇ H T )+
∂
∂z
(K V
∂T
∂z
)),
(3.20a)
F S = ρ 0 (∇ H · (K H ∇ H S)+
∂
∂z
(K V
∂S
∂z
)),
(3.20b)
with estimates K V =10 −5 – 10 −4 m 2 s −1 and K H =10– 10 3 m 2 s −1 .
Some form of friction is needed to be able to satisfy boundary conditions, for
example no-slip (zero tangential and normal velocity) conditions on the continental boundaries and on the bottom topography. In many cases, this leads to
boundary layers (e.g., the Ekman layers) whose thickness depends on the mixing
coefficients. Friction introduces characteristic time scales
τ
H
w =
L 2
A H
; τ
V
w =
D 2
A V
.
(3.21)
In vorticity terms: random transport of momentum leads to diffusion of vorticity,
with a magnitude proportional to the mixing coefficients.