Formulation of the Model
179
(4.2.4)
If the mass conservation equation is used after expanding the time
derivative in the first term in (4.2.2) we obtain for the momentum equation:
dun
~
_
1
-
-d + Jk X Un = --\7 Pn + Fn
t
Pn
(
{ iln+ 1 - iln }
) }
+ W* Zn+i)
hn
0{w*(Zn+i
+ W*(zn){iln ~~n-i }e{ -w*(zn)}.
(4.2.5)
If the cross-isopycnal mass flux at the upper interface z = Zn is positive,
there is no change produced in the momentum per unit volume of the nth layer
by motion across that interface. Momentum leaves the layer, but as it leaves,
the density of the momentum remains unchanged. (Note that the Heaviside
function is zero in this case.) On the other hand, if the cross-isopycnal flux
across this interface is negative, fluid from the layer above brings in fluid with a
different density of momentum proportional to the velocity of the n - 1st layer.
The change in momentum density is then proportional to the rate of crossisopycnal flux of the difference of the velocities of the two layers. The same
occurs at the lower interface. There is a change in momentum density only if
fluid enters the layer, in which case the change is proportional to the velocity
difference across the layers. If the cross-isopycnal mass flux is outward across
both interfaces it leads to no change, by itself, in the velocity of the layer. The
layer thickness tends to contract, and there is a consequent change in the total
momentum hniln per unit area, as described by ( 4.2.2), but not in the velocity
itself (which is the momentum density). The reader is invited now to repeat the
argument for a passive tracer (money!) to understand the dependence of the
momentum flux on the direction of the cross-isopycnal velocity. Thus, imagine
a room full of people who agree to obey the rule that each shares equally the
available cash in the room (which is equivalent to attributing the same velocity
to every element in the layer under consideration). When a rich or poor person
then enters the room with an amount of cash which differs from that which
each person already carries, the subsequent redistribution of wealth changes
the content of each person's pocketbook. If a person, however, leaves the
room, the cash possessed by each person is unaltered although the total wealth
of the room, of course, diminishes - in perfect analogy with the change in the
total momentum of our layer.
It is convenient to combine the momentum flux associated with the crossisopycnal mass flux with the friction term and write the sum as a single
dissipative term, i.e., we define:
179
(4.2.4)
If the mass conservation equation is used after expanding the time
derivative in the first term in (4.2.2) we obtain for the momentum equation:
dun
~
_
1
-
-d + Jk X Un = --\7 Pn + Fn
t
Pn
(
{ iln+ 1 - iln }
) }
+ W* Zn+i)
hn
0{w*(Zn+i
+ W*(zn){iln ~~n-i }e{ -w*(zn)}.
(4.2.5)
If the cross-isopycnal mass flux at the upper interface z = Zn is positive,
there is no change produced in the momentum per unit volume of the nth layer
by motion across that interface. Momentum leaves the layer, but as it leaves,
the density of the momentum remains unchanged. (Note that the Heaviside
function is zero in this case.) On the other hand, if the cross-isopycnal flux
across this interface is negative, fluid from the layer above brings in fluid with a
different density of momentum proportional to the velocity of the n - 1st layer.
The change in momentum density is then proportional to the rate of crossisopycnal flux of the difference of the velocities of the two layers. The same
occurs at the lower interface. There is a change in momentum density only if
fluid enters the layer, in which case the change is proportional to the velocity
difference across the layers. If the cross-isopycnal mass flux is outward across
both interfaces it leads to no change, by itself, in the velocity of the layer. The
layer thickness tends to contract, and there is a consequent change in the total
momentum hniln per unit area, as described by ( 4.2.2), but not in the velocity
itself (which is the momentum density). The reader is invited now to repeat the
argument for a passive tracer (money!) to understand the dependence of the
momentum flux on the direction of the cross-isopycnal velocity. Thus, imagine
a room full of people who agree to obey the rule that each shares equally the
available cash in the room (which is equivalent to attributing the same velocity
to every element in the layer under consideration). When a rich or poor person
then enters the room with an amount of cash which differs from that which
each person already carries, the subsequent redistribution of wealth changes
the content of each person's pocketbook. If a person, however, leaves the
room, the cash possessed by each person is unaltered although the total wealth
of the room, of course, diminishes - in perfect analogy with the change in the
total momentum of our layer.
It is convenient to combine the momentum flux associated with the crossisopycnal mass flux with the friction term and write the sum as a single
dissipative term, i.e., we define: