Acceleration of a fluid: nonlinear
dynamics
For solid masses, the description of acceleration is easy.
It’s simply the sum of all applied forces divided by the
mass.
For fluids, this definition is much more complicated
because of ADVECTION within the fluid. Advection
means that a fluid can transport properties (such as
temperature, particles, rubber duckies, etc…) according to
its velocity field.
In addition, the velocity field may also be advected by the
velocity!
An easy way to think of this is to think of an eddy in a river
being carried downstream.
The velocity measured at a point initially downstream of
the eddy will see accelerations caused by the advection of
the eddy’s velocity field by the river current.
Advection also acts on the temperature field. It is this fact
that really brings us all of our variable weather. In our part
of the world, northerly winds in winter are BAD because
they advect cold Arctic air over our location. You can think
of temperature advection in the above situation as the eddy
being a warm water eddy and the river being colder. The
temperature of the eddy is advected by the velocity of the
fluid.
In a mathematical sense, advection may be described in the
following manner. The acceleration of the fluid is divided
up into its local acceleration and its acceleration by
advection. For the zonal component of acceleration, these
are given by:
because the Sun is coming up. The latter is equivalent to
saying that
dU !U
!U
!U
!U
=
+U
+V
+W
dt
!t
!x
!y
!z
because the local temperature (ignoring advection) would
increase as radiative forcing increases.
!T
>0
!t
The first term on the right hand side is the local
acceleration. The second, third, and fourth terms are
accelerations caused by, respectively, the zonal advection
of velocity, the meridional advection of velocity, and the
vertical advection of velocity.
Note that if a fluid is at rest, these advective terms are equal
to zero so any applied force would directly accelerate the
fluid through the first term on the right. However, since it
was just accelerated it would now have a velocity field so
the advection terms would no longer be zero.
Acceleration in a fluid therefore takes the above form. The
four terms on the right hand side therefore equal all of the
applied forces (per unit mass).
TEMPERATURE ADVECTION
The concept of temperature advection can be seen quite
easily from the above terms plus the following diagram.
Imagine that the river in the previous example has a zonal
temperature gradient associated with it (perhaps there are
some nice hot springs upstream). Let’s also say that
everywhere along the river the temperature is increasing
The advection of temperature in the zonal direction is:
U
!T
!x
You should convince yourself that this is a NEGATIVE
quantity in the diagram above. In physical terms, all this
really means is that if you were floating on an inner tube on
this river, the river would carry you from warm
temperatures to cold temperatures. Therefore, advection
acts to decrease your temperature. However, since the Sun
is rising, !T/!t>0. There are competing effects, therefore, in
determining whether you warm up or cool down. If the
current is strong or the zonal temperature gradient is very
big then advection would win and you would cool down. If
the current or T gradient is weak then the Sun’s heating
would win and you would warm up.
EXAMPLE: The temperature in the river decreases by 1
degree per kilometer, and the current speed is 1 m/s. The
Sun is heating the entire area up at a rate of 10-3 degrees per
second. Do you heat up or cool down?
The advective cooling is going to be (1 m/s)(-10-3 K/m)= 10-3 K/s
This exactly balances the !T/!t=10-3 K/s heating from the
Sun. Your temperature would therefore stay the same even
though you’re being advected from a warm area to a cold
one!
Which wins? Advection or local rate of change?
Ultimately, it depends on the winds, the temperature
gradients, and the heating rates. All of these factors need to
be taken into account in numerical weather forecasting.
As a final note, the advective terms are called nonlinear
terms. They make solving the equations of motion for a
fluid very difficult indeed. In fact, only under special
circumstances can the complete fluid equations be solved
analytically (i.e., with pencil and paper). This fact was
known hundreds of years ago when the equations were first
derived.
Now, however, we have computers that can solve them
approximately using a variety of numerical techniques.
This fact has led to the entire science behind weather and
climate forecasting. It also led to the discovery of chaos
theory (have you heard of the butterfly flapping its wings?)
and a whole branch of dynamical systems theory.