Mathematical description
69
(3.4) Equations of motion
It is a useful exercise to derive the equations of motion (3.32a-c) from the
specification of the Navier-Stokes equations in spherical coordinates (r, ϑ, ϕ)
such as provided, for example in appendix 2 of Batchelor (2000).
First consider the isotropic case for which A H = A V .
a. Give an explicit expression for the inertial terms in (3.32a).
b. Give an explicit expression for the term F
φ
I in (3.32a).
Next consider the nonisotropic case, but now in Cartesian coordinates.
c. Give an explicit expression for the term F
φ
I in (3.32a) using (3.18).
(3.5) Damped inertial motion
We consider the situation where a particular wind stress has driven a flow in
an ocean basin (containing water of constant density) for a while and then
suddenly ceases. At this point there are no external forces acting on the water.
There is a linear friction damping the motion with a friction coefficient r.T h e
horizontal momentum equations are
Du
dt
=2 Ω v sin φ − ru
Dv
dt
= −2Ωu sin φ − rv
where D/dt is the material derivative. Assume that a water parcel has a
horizontal velocity (u, v)=(0,v 0 ) at t =0.
a. Show that
D
dt
(u
2 + v
2 )=−2r(u
2 + v
2 )
We now search for solutions of the form
(u(t),v(t)) = e
−rt (c 1 sin(αt +Ψ 1 ),c 2 cos(αt +Ψ 2 )).
with constants α, c 1 , c 2 and Ψ 2 .
b. Determine (u(t),v(t)) and explain what kind of motion of the water parcel
results.