Mathematical description
65
3.3. Dominant balances
We return to the general equations (3.32) and ask the question whether we can
determine aprioridominant balances of terms in the these equations. Therefore
it is useful to consider the stationary solution of these equations when the water
is motionless. After substitution of v =0into (3.32), we find that this solution
(indicated by ¯
p, ¯
T, ¯
S, ¯
ρ) must satisfy (with z = r − r 0 )
∂ ¯
p
∂φ
=
∂ ¯
p
∂θ
=0 ,
(3.40a)
∂ ¯
p
∂z
= −¯ ρg,
(3.40b)
F T +
Q T
¯
ρC p
=0 ,
(3.40c)
F S +
Q S
¯
ρ
=0 ,
(3.40d)
¯
ρ − ρ(¯ p, ¯
T, ¯
S)=0 ,
(3.40e)
where the fact that there is no mixing of momentum when v =0has been used.
If Q S = Q T =0 ,then ¯
T and ¯
S are only functions of z which are determined by
the boundary conditions. The density ¯
ρ(z) is calculated from (3.40e) and (3.40b)
and determines the hydrostatic pressure ¯
p(z).
Next we consider deviations from this hydrostatic steady state and introduce
the dynamic pressure ˜
p and density ˜
ρ such that p =¯ p +˜ p and ρ =¯ ρ +˜ ρ.T o
Ex. 3.5
estimate the magnitude of these dynamic quantities, we use the time scales τ a ,
τ f and τ w . For flows with a horizontal length scale L and a horizontal velocity
scale U , we first consider the horizontal momentum balances. The magnitude
of the inertial accelerations ρv.∇v can be estimated as ρ 0 U 2 /L = ρ 0 U/τ a and
that of the Coriolis accelerations as 2Ωρ 0 U sin θ = ρ 0 U/τ f . As the time scale
τ w is much larger than both τ a and τ f and τ f ≪ τ a , the dominant balance must
be between Coriolis acceleration and pressure gradient. But because the Coriolis
acceleration depends on the latitude θ, we have to consider three different cases:
(i) Midlatitude ocean circulation
The flow can be considered near a latitude θ 0 =0and it is local so that L/r 0 is
small. The scales for dynamic pressure and density are determined in chapters
5 (homogeneous case) and 7 (stratified case) and the governing equations are
reduced according to these scales.
(ii) Equatorial ocean circulation
The flow can be considered locally near the latitude θ 0 ≈ 0 and L/r 0 is small.
In this case, the Coriolis acceleration at the equator is zero, but it is nonzero
65
3.3. Dominant balances
We return to the general equations (3.32) and ask the question whether we can
determine aprioridominant balances of terms in the these equations. Therefore
it is useful to consider the stationary solution of these equations when the water
is motionless. After substitution of v =0into (3.32), we find that this solution
(indicated by ¯
p, ¯
T, ¯
S, ¯
ρ) must satisfy (with z = r − r 0 )
∂ ¯
p
∂φ
=
∂ ¯
p
∂θ
=0 ,
(3.40a)
∂ ¯
p
∂z
= −¯ ρg,
(3.40b)
F T +
Q T
¯
ρC p
=0 ,
(3.40c)
F S +
Q S
¯
ρ
=0 ,
(3.40d)
¯
ρ − ρ(¯ p, ¯
T, ¯
S)=0 ,
(3.40e)
where the fact that there is no mixing of momentum when v =0has been used.
If Q S = Q T =0 ,then ¯
T and ¯
S are only functions of z which are determined by
the boundary conditions. The density ¯
ρ(z) is calculated from (3.40e) and (3.40b)
and determines the hydrostatic pressure ¯
p(z).
Next we consider deviations from this hydrostatic steady state and introduce
the dynamic pressure ˜
p and density ˜
ρ such that p =¯ p +˜ p and ρ =¯ ρ +˜ ρ.T o
Ex. 3.5
estimate the magnitude of these dynamic quantities, we use the time scales τ a ,
τ f and τ w . For flows with a horizontal length scale L and a horizontal velocity
scale U , we first consider the horizontal momentum balances. The magnitude
of the inertial accelerations ρv.∇v can be estimated as ρ 0 U 2 /L = ρ 0 U/τ a and
that of the Coriolis accelerations as 2Ωρ 0 U sin θ = ρ 0 U/τ f . As the time scale
τ w is much larger than both τ a and τ f and τ f ≪ τ a , the dominant balance must
be between Coriolis acceleration and pressure gradient. But because the Coriolis
acceleration depends on the latitude θ, we have to consider three different cases:
(i) Midlatitude ocean circulation
The flow can be considered near a latitude θ 0 =0and it is local so that L/r 0 is
small. The scales for dynamic pressure and density are determined in chapters
5 (homogeneous case) and 7 (stratified case) and the governing equations are
reduced according to these scales.
(ii) Equatorial ocean circulation
The flow can be considered locally near the latitude θ 0 ≈ 0 and L/r 0 is small.
In this case, the Coriolis acceleration at the equator is zero, but it is nonzero