3.7 Forces
29
3.7.6 Interpretation of the Advection Equation
The existence of both a f ow and temperature gradients are essential ingredients in
the advection process. The left-hand side of (3.7) is the temporal change in temperature measured at a f xed location. The appearance of minus signs on the right-hand
side of (3.7) is not that difficul to understand. For simplicity, consider a f ow running
parallel to the x-direction. Recall that, per definition u is positive if this fl w component runs into the positive x-direction. Warming over time (∂ T /∂t > 0) occurs
with an increase of T in the x-direction in conjunction with a negative u. Warming
also occurs with a positive u but a decrease of T in the x-direction. I am sure that
the reader can work out scenarios leading to a local cooling.
In the absence of either fl w or temperature gradients, Eq. (3.7) turns into:
∂ T
∂t
= 0
(3.8)
This equation simply means that temperature does not show changes at a certain
location. The important difference with respect to (3.6) is that this relation holds for
a f xed location, whereas the other one was for an observer moving with the f ow.
Most ocean models use the Eulerian frame of reference.
3.7.7 The Nonlinear Terms
Flow can advect different properties such as gradients in temperature, salinity and
nutrients, but also momentum; that is, the components of velocity itself. The resultant terms are called the nonlinear terms. These terms are included in Newton’s
second law of motion, if we express this in an Eulerian frame of reference, yielding:
∂u
∂t
+ u
∂u
∂ x
+ v
∂u
∂ y
+ w
∂u
∂z
=
m
i=1
F
x
i
∂v
∂t
+ u
∂v
∂ x
+ v
∂v
∂ y
+ w
∂v
∂z
=
m
i=1
F
y
i
(3.9)
∂w
∂t
+ u
∂w
∂ x
+ v
∂w
∂ y
+ w
∂w
∂z
=
m
i=1
F
z
i
The nonlinear terms are traditionally written on the left-hand side of the momentum conservation equations for they are no true forces.
29
3.7.6 Interpretation of the Advection Equation
The existence of both a f ow and temperature gradients are essential ingredients in
the advection process. The left-hand side of (3.7) is the temporal change in temperature measured at a f xed location. The appearance of minus signs on the right-hand
side of (3.7) is not that difficul to understand. For simplicity, consider a f ow running
parallel to the x-direction. Recall that, per definition u is positive if this fl w component runs into the positive x-direction. Warming over time (∂ T /∂t > 0) occurs
with an increase of T in the x-direction in conjunction with a negative u. Warming
also occurs with a positive u but a decrease of T in the x-direction. I am sure that
the reader can work out scenarios leading to a local cooling.
In the absence of either fl w or temperature gradients, Eq. (3.7) turns into:
∂ T
∂t
= 0
(3.8)
This equation simply means that temperature does not show changes at a certain
location. The important difference with respect to (3.6) is that this relation holds for
a f xed location, whereas the other one was for an observer moving with the f ow.
Most ocean models use the Eulerian frame of reference.
3.7.7 The Nonlinear Terms
Flow can advect different properties such as gradients in temperature, salinity and
nutrients, but also momentum; that is, the components of velocity itself. The resultant terms are called the nonlinear terms. These terms are included in Newton’s
second law of motion, if we express this in an Eulerian frame of reference, yielding:
∂u
∂t
+ u
∂u
∂ x
+ v
∂u
∂ y
+ w
∂u
∂z
=
m
i=1
F
x
i
∂v
∂t
+ u
∂v
∂ x
+ v
∂v
∂ y
+ w
∂v
∂z
=
m
i=1
F
y
i
(3.9)
∂w
∂t
+ u
∂w
∂ x
+ v
∂w
∂ y
+ w
∂w
∂z
=
m
i=1
F
z
i
The nonlinear terms are traditionally written on the left-hand side of the momentum conservation equations for they are no true forces.