Dynamics of the Oceans and Atmosphere
11:670:324
Spring 2011
The final exam will be comprehensive, including the following material:
1) Expansion of total derivatives in height and pressure coordinates
2) Evaluating sign of derivatives and wind components

Dynamics of the Atmosphere
11:670:324
Spring 2011
Additional material to know for second hourly exam
1) Implications of chaos for numerical weather prediction
2) Pressure coordinates; momentum equation in pressure coordinates; characteristics
of pressure

Dynamics of the Atmosphere
11:670:324
Spring 2011
Material to know for first hourly exam:
1) Expansion of total derivatives
2) Evaluating sign of derivatives and wind components
3) Decomposition of vectors into components
4) Vector operations, especially

Name_
Dynamics of the Atmosphere 11:670:324
Spring 2010
Dr. Anthony J. Broccoli
Final Exam
May 11, 2010
Unless otherwise noted, please answer each of the following questions in the blue book. Read
each question carefully before formulating your answer. Sh

Name_
Dynamics of the Atmosphere 11:670:324
Spring 2010
Dr. Anthony J. Broccoli
Hourly Exam #2
April 9, 2010
Unless otherwise noted, please answer each of the following questions in the blue book. Read
each question very carefully before formulating your

Name_
Dynamics of the Oceans and Atmosphere 11:670:324
Spring 2008
Dr. Anthony J. Broccoli
Hourly Exam #1
March 7, 2008
Unless otherwise noted, please answer each of the following questions in the blue book. Read
each question very carefully before formul

Vorticity Equation in Isobaric Coordinates
To obtain a version of the vorticity equation in pressure coordinates,
we follow the same procedure as we used to obtain the z-coordinate
version:
[y-component momentum equation]
[x-component momentum equation]

The Vorticity Equation
To understand the processes that produce changes in vorticity, we
would like to derive an expression that includes the time derivative of
vorticity:
d v u
=K
dt x y
Recall that the momentum equations are of the form
du
=K
dt
dv
=

Vorticity
Vorticity is the microscopic measure of spin and rotation in a fluid.
Vorticity is defined as the curl of the velocity:
r
V
Wind direction varies clockwise spin
Wind speed varies clockwise spin
Absolute vorticity (inertial reference frame):
Rel

Thermal Wind Equation
We begin by writing the vector form of the geostrophic wind equation
in isobaric coordinates for two levels:
(V ) = g ( Z ) k
f
r
g1
(V )
r
g2
p
=
1
(
)
g
pZ 2 k
f
Now we compute the difference in geostrophic wind between the two le

Balanced Flow
The pressure and velocity distributions in atmospheric
systems are related by relatively simple, approximate
force balances.
We can gain a qualitative understanding by considering
steady-state conditions, in which the fluid flow does not
v

Hydrostatic Balance
dp
= g
dz
In the absence of atmospheric motions the gravity force
must be exactly balanced by the vertical component of
the pressure gradient force.
Because vertical accelerations are very small for largescale atmospheric motions, th

Primitive Equations
du
1 p
fv =
x
dt
dv
1 p
+ fu =
y
dt
dp
= g
dz
1 d
u v w
+
=
dt
x y z
dT
dp
cp
=Q
dt
dt
p = RT
x-component momentum equation
y-component momentum equation
hydrostatic equation
continuity equation
thermodynamic energy equation
equa

Conservation of Energy
For a system in thermodynamic equilibrium, the first law
of thermodynamics states that the change in internal
energy of the system is equal to the difference between
the heat added to the system and the work done by the
system.
Th

Are All Of These Terms Important?
du uv tan uw
1 p
+ 2v sin 2w cos + Frx
+
=
dt
a
a
x
dv u 2 tan vw
1 p
+
+
=
2u sin + Fry
dt
a
a
y
dw u 2 + v 2
1 p
=
+ 2u cos g + Frz
dt
a
z
Scale Analysis
Goal: To determine relative importance of the
terms in the b

Momentum Equations in Spherical
Coordinates
For a variety of reasons, it is useful to express
the vector momentum equation for a rotating
earth as a set of scalar component equations.
The use of latitude-longitude coordinates to
describe positions on ea

Total Differentiation of a Vector in a
Rotating Frame of Reference
Before we can write Newtons second law of motion for a
reference frame rotating with the earth, we need to
develop a relationship between the total derivative of a
vector in an inertial r

Frames of Reference
Newtons laws of motion are valid in a coordinate system
that is fixed in space.
A coordinate system fixed in space is known as an
inertial (or absolute) frame of reference.
A coordinate system that is not fixed in space, such as
one

The Atmospheric Continuum
In atmospheric dynamics (ocean dynamics also) we do
not treat the fluid as a collection of individual molecules.
Instead we treat the fluid as a continuous medium (or
continuum) in which a point is a volume element that
is very

Kinematics
The atmosphere is
characterized by horizontal
(and vertical) variations in
the wind field.
For example, look at the
horizontal variations of wind
speed and direction (and
hence u and v) on this
weather map depicting
surface streamlines.
These h

Scale Analysis
Your job is to fill an empty swimming pool and
you have been asked how long it will take.
You can write an equation that takes all of the
inflows and outflows into account, but you do
not have exact values for any of them.
Does that mean

Vectors and Vector Analysis
Scalar Representation of a
quantity using only a number
(e.g., temperature).
Vector Representation of a
quantity that has both
magnitude and direction (e.g.,
wind velocity).
Unit Vectors Vectors with unit
length (i.e., magni

Coordinate Systems
To describe the
location in space of a
point in a fluid, a
coordinate system is
used.
A commonly used
coordinate system is
the rectangular, or
x,y,z system (also
known as Cartesian).
z
(x0,y0,z0 )
y
z0
x0
y0
x
Rectangular coordinates