1

Total Derivative

Suppose . The differential of this is .

If we divide this expression by the dt we obtain

where

=total derivative (Lagrangian method of describing the motion); for changes in following the parcel,

=local derivative (Eulerian method of describing the motion). Changes in are at a fixed point in space (holdingx, y, and z constant).

Using vector notation,

or if you want an equation for describing the change of at a fixed point then,

.

Suppose was temperature. The equation above would describe the change of temperature at a fixed point.

If , we say that the field is in steady state. If , we say that is a conservative quantity (i.e., the temperature doesnʼt change following the parcel).

In this particular example where we assume is temperature, real examples of effects that cause , are the release of LATENT HEAT OR RADIATION EFFECTS. For example, if a parcel of air was being warmed by the sun as it was moving, . 2

is an important effect. This is referred to as the advective term. It states that the temperature can change if warmer or colder air is advected into a region. It is critical to note that there is a minus sign in front of this term. Letʼs check this term out with a simple example.

y

θ4 θ3 θ2 θ1 θ4 >θ3 > θ2 > θ1

warm air cold air

UCLA

west wind (u>0, v=w=0)

x

We assume that (no radiation or latent heat effects) and ask the question - how will the temperature change at UCLA with increasing time?

We know that the temperature will rise at UCLA just by looking at the figure, but letʼs see if the equation predicts the same answer.

. assume 0 v=0 w=0

We know that u>0 and since the temperature is decreasing in the positive x-direction. ∂θ > 0 Therefore, ∂t . Again, remember that there is a minus sign in front of the advective terms. 3

VORTICITY

Vorticity is a vector measure of the tendency of a fluid parcel to rotate about an axis through its center of mass. Vorticity = .

All of the components of the vorticity vector are important. However, for large-scale, synoptic flow we tend to be primarily interested in the tendency for parcels to rotate about their vertical axis.

>0 with counterclockwise rotation (cyclonic in the northern hemisphere).

<0 with clockwise rotation (anticyclonic in the northern hemisphere).

Rather than use a Cartesian coordinate system sometimes it is beneficial to use the “natural coordinate system” to describe certain charteristics of the flow (see Holtonʼs book).

In short, the natural coordinates assume the following,

and

.

We no longer have a fixed coordinate system when we use natural coordinates. The natural coordinates are defined by the velocity vector. The -direction always points in the same direction as the velocity vector, and the -direction always points normal and to the left of the velocity vector.

At first glance, this seems to be a more difficult system to work in. As the velocity vector changes its orientation, the and directions are always changing. However, you will see that it makes “interpretation” simpler. 4

It can be shown that vertical vorticity in natural coordinates can be written as,

where R=radius of curvature of the flow.

is often referred to as the curvature term and is referred to as the shear term.

This new way at looking at the flow tells us that vertical vorticity can be created by both curved flows and by speed shear. The former is straightforward, but the latter may be illustrated by the following:

n

t We see in the above example that the flow has no curvature effect. We also note thatV is decreasing in the direction. Therefore,

.

This should produce cyclonic vorticity, which is confirmed by our expectations that the leaf will rotate counterclockwise if it was embedded in this flow. 5

DIVERGENCE

Divergence of a velocity field can be represented by

.

Divergence is a measure of the tendency of a vector field to expand or spread out. It is a scalar field. For large scale flows, we are often only interested in the horizontal divergence or,

.

Once again, natural coordinates are a useful way to understand what contributes to divergence. It can be shown that

where the first term measures the change of speed in the direction and the second term measures how the wind direction is changing in the direction (winds turning counterclockwise in the increasing direction means is ). 6

The following schematics will help you understand these two terms: n ∂V > 0 , divergent flow t ∂s

n ∂V < 0 , convergent flow t ∂s

∂θ θ V > 0 , divergent flow. This effect is also ∂n referred to as difluence.

∂θ V < 0 , convergent flow. This effect is also ∂n referred to as confluence

θ This is an example where streamlines are diverging but the flow is slowing down. Accordingly, the flow may be non-divergent. This shows that difluent flow may not be truly divergent. 7

Here is an example of a horizontal flow that is convergent.