GEF 2500 Problem set 5
1) Explain how air temperature affects the vertical pressure gradient.
2) For a vase filled with homogeneous water under the influence of gravity, what is the
pressure near the bottom directly under the hole at the top at depth H (point A)? Now
this vase, like most bottles, is wider near the bottom than at the top. What is the
pressure near the side of the vase, still at depth H, but in a location that is not under the
hole at the top (location B)? Explain!
H
A vase filled with water
A
B
3) (ex. 2.5 in Vallis)
A fluid at rest evidently satisfies the hydrostatic relation, which says that the pressure
at the surface is given by the weight of the fluid above it. Now consider a deep
atmosphere on a spherical planet. A unit cross-sectional area at the planet’s surface
lies beneath the column of fluid whose cross-section increases with height, because the
total area of the atmosphere increases with distance away from the center of the planet.
Is the pressure at the surface still given by the hydrostatic relation, or is it greater than
this because of the increased mass of the fluid in the column? If it is still given by the
hydrostatic relation, then the pressure at the surface, integrated over the entire area of
the planet, is less than the total weight of the fluid; resolve this paradox. But if the
pressure at the surface is greater than that implied by hydrostatic balance, explain how
the hydrostatic relation fails.
cross-section
Atmosphere
Planet
Hint: The radial component of the momentum equation is
4) Inertial Oscillation
Consider a two-dimensional, inviscid fluid flow in a rotating frame of reference on the f-plane
(i.e. set f constant). Assume that the speed of fluid parcels is constant in space.
a) Ignore the pressure term. Write down the horizontal momentum equations and show
that when given the assumptions above the equations are reduced to:
b)
Determine the general solution to equations 1-2. Let the initial conditions be
u(t=0)=u0 and v(t=0)=0, where
is the horizontal velocity vector. Show
that the trajectory of the fluid parcel is a circle with radius |u0|/f, where |u0| is the fluid
speed.
c) What is the period of oscillation of a fluid parcel located at the North Pole?
Inasmuch as the Coriolis force is the only force in equations 1, it is tempting to assume that
the Coriolis force is responsible for the inertial oscillation. Indeed the name inertial oscillation
suggests that the motion arises solely as a result of inertial forces, which are apparent forces
that appear in accelerating coordinate frames (e.g. Coriolis and centrifugal forces). In fact,
inertial oscillations are not produced entirely by inertial forces, and even viewed from a nonrotating coordinate frame, the inertial oscillation looks like oscillatory motion. One external
force plays an essential role in driving the inertial oscillation. (from D. Durran 1993)
The inertial oscillation as viewed from a non-rotating reference frame
In the following problems we will try to describe the oscillation that would be seen by an
observer in a non-rotating coordinate frame. Consider, therefore, a f-plane tangent to the
earth at the North Pole. Defining the angular velocity vector as
equations 1-2 can be written as a single vector equation
d) Start with equation (1.1.1) in the compendium. Since we are at the North Pole, ⃗
and find the equation governing the inertial oscillation in the fixed coordinate frame,
i.e.
⃗
where vf is the horizontal velocity vector in the fixed coordinate system.
(Hint: use eqn. (1.1.1) and eqn. 4)
e) The right hand side of the equation governing the inertial oscillation in the fixed
coordinate frame (i.e. eqn. 5) is not zero and must represent a real external force
since the equation describes motion in a non-rotating coordinate frame. What kind of
force? Explain!
f)
If the origin coincides with the North Pole, eqn 5, may be expressed in component
form as:
Assume that the parcel is located at the North Pole at t=0. Determine the motion of an
air parcel with respect to the non-rotating coordinate system. The initial velocities are
as before u(t=0)=u0 and v(t=0)=0.
g) What is the period of oscillation of a fluid parcel in the non-rotating coordinate system?
h)
Make a sketch where you describe the relative position of the fixed- and rotating
coordinate trajectories for t=0, 3, 6, 9, 12 and 24 hours. Indicate the location of the
north pole and the position of an air-parcel with initial position at the north pole at t=0.
If you want you can make a Matlab or a Python program.