GEF3450
Exercises for group session: Streamfunctions
Ada Gjermundsen
E-mail: ada.gjermundsen@geo.uio.no
November 2, 2015
Exercises from “Atmosphere, Ocean and Climate Dynamics” by Marshall and Plumb and
“Atmospheric Science” by Wallace and Hobbs.
The streamfunction ψ is defined for incompressible (divergence-free) flows in
two dimensions. The flow velocity components can then be expressed as
u=−
∂ψ
∂y
v=
∂ψ
∂x
(1)
The streamfunction can be used to plot streamlines, which represent the
trajectories of particles in a steady flow.
1 Define a streamfunction ψ for a non-divergent, two-dimensional flow
in a vertical plane:
∂u ∂v
+
=0
∂x ∂y
(2)
and interpret it physically.
Show that the instantaneous particle paths (streamlines) are defined by
ψ = const, and hence in steady flow the contours ψ = const are particle
trajectories. When are trajectories and streamlines not coincident?
2 A two-dimensional steady flow has velocity components
u=y
v=x
(3)
show that the streamlines are rectangular hyperbolas
x2 − y 2 = const.
(4)
Sketch the flow pattern, and convince yourself that it represents an
irrotational flow in a 90◦ corner.
1
3 For streamfunctions ψ with the following functional forms, sketch the
velocity field:
i) ψ = my
ii) ψ = my + n cos(2πx/L)
iii) ψ = m(x2 + y 2 )
iv) ψ = myx
where m and n are constants.
4 For each of the flows in the previous exercise
i) Calculate the strain and rotation matrices
ii) Describe the distribution of vorticity
5 Consider a streamfunction:
ψ = x3 + y 3
(5)
i) Sketch the streamlines.
ii) What are the strain and rotation matrices?
iii) Describe what would happen to a square element advected by this
flow. Will it conserve its area?
6 Consider a velocity field that can be represented as
~ = −k̂ × ∇ψ
V
(6)
~ is everywhere
where ψ is called the streamfunction. Prove that ∇H · V
equal to zero and that the vorticity field is given by
ζ = ∇2 ψ
2
(7)
Some solutions:
3 i) a pure zonal flow: eastward if m < 0 and westward if m > 0
ii) a wavy zonal flow in which the wavelengths of the wave is L
iii) solid body rotation: counterclockwise if m > 0
iv) deformation: if m > 0 the y-axis is the axis of streaching, if
m < 0 the x-axis is the axis of streaching
4b i) ζ = 0 everywhere
ii) ζ = (2π/L)2 n cos(2πx/L), sinusoidal in x, independent of y
iii)ζ = 4m everywhere
iv) ζ = 0 everywhere
3