Thermocline problem
309
in Fig. 13.2, then a surface density distribution can be mimicked. In this way, the
vertical density difference is also coupled to the horizontal density difference.
13.5. Thermocline theory
There are two different theories of the physics of the thermocline: the ventilation theory and the internal boundary layer theory. Both can considered as limits
of a final theory, which unfortunately has not been developed yet. The ventilation
theory has the attractive property that it is purely advective. However, it can only
provide a good description in cases where the surface Ekman pumping velocity
ˆ
w E < 0. The internal boundary layer theory includes a dependency on the magnitude of λ V in (13.3). To give realistic thermocline depths the value of λ V has to
be larger than indicated by observations.
13.5.1. Ventilation theory
The central idea of the ventilation theory is that properties of the surface ocean
are transported through advection to deeper regions of the ocean. In other words,
these properties are ventilated from the surface to the deeper ocean. In this way,
water of relatively high density in polar areas can be advected equatorward (southward in the northern hemisphere) below surface water which has a smaller density.
The net result is the formation of a thermocline. The ventilation theory is purely
advective as with λ V =0in (13.3) we find
Dρ
dt
=0.
(13.49)
and the density is constant along streamlines. As one can anticipate problems
will arise to satisfy surface boundary conditions for the density when the Ekman
vertical velocity is negative.
We consider the ventilation process in more detail using a simple three-layer
ocean model, as sketched in Fig. 13.3. The flow domain is bounded by a flat
bottom and coastlines at φ = φ W and φ = φ E . The layers have a constant
density ρ j and the third layer is assumed to be motionless. The location of the
interfaces between the layers are indicated by the dimensionless z-coordinates z j ,
with z 1 =0, z 4 = −1,and
z 2 = −h 1 ,
(13.50a)
z 3 = −(h 1 + h 2 ).
(13.50b)
The effect of ocean-atmosphere deformation can be neglected on the large
scale, and the layer thicknesses are given by
h 1 = −z 2 ,
(13.51a)
h 2 = z 2 − z 3 ,
(13.51b)
h 3 = z 3 +1.
(13.51c)
309
in Fig. 13.2, then a surface density distribution can be mimicked. In this way, the
vertical density difference is also coupled to the horizontal density difference.
13.5. Thermocline theory
There are two different theories of the physics of the thermocline: the ventilation theory and the internal boundary layer theory. Both can considered as limits
of a final theory, which unfortunately has not been developed yet. The ventilation
theory has the attractive property that it is purely advective. However, it can only
provide a good description in cases where the surface Ekman pumping velocity
ˆ
w E < 0. The internal boundary layer theory includes a dependency on the magnitude of λ V in (13.3). To give realistic thermocline depths the value of λ V has to
be larger than indicated by observations.
13.5.1. Ventilation theory
The central idea of the ventilation theory is that properties of the surface ocean
are transported through advection to deeper regions of the ocean. In other words,
these properties are ventilated from the surface to the deeper ocean. In this way,
water of relatively high density in polar areas can be advected equatorward (southward in the northern hemisphere) below surface water which has a smaller density.
The net result is the formation of a thermocline. The ventilation theory is purely
advective as with λ V =0in (13.3) we find
Dρ
dt
=0.
(13.49)
and the density is constant along streamlines. As one can anticipate problems
will arise to satisfy surface boundary conditions for the density when the Ekman
vertical velocity is negative.
We consider the ventilation process in more detail using a simple three-layer
ocean model, as sketched in Fig. 13.3. The flow domain is bounded by a flat
bottom and coastlines at φ = φ W and φ = φ E . The layers have a constant
density ρ j and the third layer is assumed to be motionless. The location of the
interfaces between the layers are indicated by the dimensionless z-coordinates z j ,
with z 1 =0, z 4 = −1,and
z 2 = −h 1 ,
(13.50a)
z 3 = −(h 1 + h 2 ).
(13.50b)
The effect of ocean-atmosphere deformation can be neglected on the large
scale, and the layer thicknesses are given by
h 1 = −z 2 ,
(13.51a)
h 2 = z 2 − z 3 ,
(13.51b)
h 3 = z 3 +1.
(13.51c)