Dynamics of ENSO
279
where ∇ 2
H is the horizontal Laplace operator. The horizontal and vertical diffusivities are indicated by K H and K V (both m 2 s −1 ), respectively. The boundary
conditions are
z ∗ =0 : ρC p K V
∂T ∗
∂z ∗
= Q oa ,
(12.3a)
z ∗ = −H m : ρC p K V
∂T ∗
∂z ∗
= Q b .
(12.3b)
where ρ is the constant density in the mixed layer and C p the constant heat capacity. Using the approximation that the temperature is vertically homogeneous over
the layer, one can integrate (12.2) over the layer which results in
∂T ∗
∂t
+ u ∗
∂T ∗
∂x ∗
+ v ∗
∂T ∗
∂y ∗
= K H ∇
2
H T ∗ +
Q oa − Q b
ρC p H m
.
(12.4)
The net heat flux Q oa can be parameterized into a simple form as, for example,
given by
Q oa = −a 1 (T ∗ − T r∗ ),
(12.5)
where T r∗ is a reference atmospheric equilibrium temperature. At the lower
boundary, the heat flux is composed of diffusive and advective contributions, with
the latter dominating. An approximation of this heat flux is
Q b
ρC p
= w ∗
T ∗ − T s∗
H u
,
(12.6)
where w ∗ is a typical vertical velocity at the bottom of the mixed layer, H u a
vertical distance such that the temperature gradient between the mixed layer and
the subsurface temperature T s∗ is well approximated. The subsurface temperature
will depend on the vertical temperature distribution and hence on the position of
the thermocline. For example, if the thermocline depth increases then the colder
water is further from the surface and T s∗ will increase.
12.2.2. Wind induced ocean flow anomalies
Ex. 12.1
As can be deduced from Fig. 12.4, SST anomalies lead to surface wind-stress
anomalies. From the theory on the equatorial ocean circulation in chapter 11,
we know that wind-stress anomalies lead to (i) changes in the Ekman flow, (ii)
changes in the thermocline slope and (iii) the generation of a spectrum of free
waves, in particular of Kelvin and Rossby waves.
The surface Ekman layer theory was discussed in section 11.3 using a Laplacian form of momentum friction. In the reduced gravity model in section 11.2,
however, linear friction was used. If the frictional processes in the equatorial Ekman layer are idealized to be linear, with damping coefficient a s , then a vertically
279
where ∇ 2
H is the horizontal Laplace operator. The horizontal and vertical diffusivities are indicated by K H and K V (both m 2 s −1 ), respectively. The boundary
conditions are
z ∗ =0 : ρC p K V
∂T ∗
∂z ∗
= Q oa ,
(12.3a)
z ∗ = −H m : ρC p K V
∂T ∗
∂z ∗
= Q b .
(12.3b)
where ρ is the constant density in the mixed layer and C p the constant heat capacity. Using the approximation that the temperature is vertically homogeneous over
the layer, one can integrate (12.2) over the layer which results in
∂T ∗
∂t
+ u ∗
∂T ∗
∂x ∗
+ v ∗
∂T ∗
∂y ∗
= K H ∇
2
H T ∗ +
Q oa − Q b
ρC p H m
.
(12.4)
The net heat flux Q oa can be parameterized into a simple form as, for example,
given by
Q oa = −a 1 (T ∗ − T r∗ ),
(12.5)
where T r∗ is a reference atmospheric equilibrium temperature. At the lower
boundary, the heat flux is composed of diffusive and advective contributions, with
the latter dominating. An approximation of this heat flux is
Q b
ρC p
= w ∗
T ∗ − T s∗
H u
,
(12.6)
where w ∗ is a typical vertical velocity at the bottom of the mixed layer, H u a
vertical distance such that the temperature gradient between the mixed layer and
the subsurface temperature T s∗ is well approximated. The subsurface temperature
will depend on the vertical temperature distribution and hence on the position of
the thermocline. For example, if the thermocline depth increases then the colder
water is further from the surface and T s∗ will increase.
12.2.2. Wind induced ocean flow anomalies
Ex. 12.1
As can be deduced from Fig. 12.4, SST anomalies lead to surface wind-stress
anomalies. From the theory on the equatorial ocean circulation in chapter 11,
we know that wind-stress anomalies lead to (i) changes in the Ekman flow, (ii)
changes in the thermocline slope and (iii) the generation of a spectrum of free
waves, in particular of Kelvin and Rossby waves.
The surface Ekman layer theory was discussed in section 11.3 using a Laplacian form of momentum friction. In the reduced gravity model in section 11.2,
however, linear friction was used. If the frictional processes in the equatorial Ekman layer are idealized to be linear, with damping coefficient a s , then a vertically