Exercise, GEF2500 1. What is the Bousinesq-approximation?

Exercise, GEF2500
1. What is the Bousinesq-approximation?
2. Let the x- and y-axis be horizontal, and the z-axis positive upwards. Use the Boussinesq-
approximation and asume that the horizontal velocities v H only vary with z. Write down the
equations that control the mean motion in component form. Clarify why some terms may be
3. We neglect the friction and let the ocean surface be given by z=0. The density is
ρ = ρ r (1 + ay − bz )
where a,b are constant and greater than zero, and ρ r is a reference density. The pressure at the
surface p(z=0) = p0 is constant.
a) Show that the vertical velocity w is zero everywhere if it is zero at the surface.
b) Find p and derive ∇p × ∇ρ . Sketch how p and ρ vary in a cross-section in the yz-plane.
c) Find v H . What is this type of motion called?
4. Derive the Brunt-Väisälä-frequency N from the definition
N2 = −
g ∂ρ
ρ r ∂z
Let g=9.8 m s-2, and assume that the internal oscillation period is 10 minutes. Determine b.