Wind-driven circulation
125
b. Give a physical explanation of the upwelling velocity in the northern
hemisphere, in case τ 0 < 0.
c. Introduce a coordinate λ =(x E − x)/δ and derive that the dominant zonal
momentum balance is given by
u +(
A H
f
)
2 u xxxx ≈ 0
d. Determine the lateral Ekman boundary layer scale δ.
(5.5) Steady geostrophic flow
Consider a thin layer of water that rotates with a constant angular velocity Ω;
the layer is bounded from below by a flat bottom. Initially, the water is motionless and the deviation h of the dimensionless sea surface from its equilibrium
value is zero. At t =0, the sea surface is deformed according to
h(x, y)=(−H(x)+H(−x))h 0
where h 0 is a constant and H is the Heaviside function. During the development of the flow, the Rossby number ǫ remains small.
a. Use quasi-geostrophic theory and formulate the equations which determine
the geostrophic velocities u 0 , v 0 and surface deformation h 0 .
Assume now that the amplitudes of velocities are so small that products of
these quantities can be neglected.
b. Determine the linear equation describing the evolution of h 0 .
Consider now the special case that the steady state field h 0 does not depend
on y.
c. Determine the steady state field h 0 .
d. Determine and sketch the geostrophic velocities u 0 and v 0 .
125
b. Give a physical explanation of the upwelling velocity in the northern
hemisphere, in case τ 0 < 0.
c. Introduce a coordinate λ =(x E − x)/δ and derive that the dominant zonal
momentum balance is given by
u +(
A H
f
)
2 u xxxx ≈ 0
d. Determine the lateral Ekman boundary layer scale δ.
(5.5) Steady geostrophic flow
Consider a thin layer of water that rotates with a constant angular velocity Ω;
the layer is bounded from below by a flat bottom. Initially, the water is motionless and the deviation h of the dimensionless sea surface from its equilibrium
value is zero. At t =0, the sea surface is deformed according to
h(x, y)=(−H(x)+H(−x))h 0
where h 0 is a constant and H is the Heaviside function. During the development of the flow, the Rossby number ǫ remains small.
a. Use quasi-geostrophic theory and formulate the equations which determine
the geostrophic velocities u 0 , v 0 and surface deformation h 0 .
Assume now that the amplitudes of velocities are so small that products of
these quantities can be neglected.
b. Determine the linear equation describing the evolution of h 0 .
Consider now the special case that the steady state field h 0 does not depend
on y.
c. Determine the steady state field h 0 .
d. Determine and sketch the geostrophic velocities u 0 and v 0 .