MKEP4 Exercise 05 WS1819 PDF

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Wintersemester 2018/19, Prof. Dr. Aeschbach-Hertig, Prof. Dr. André Butz...

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MKEP4 - Master Course Environmental Physics winter term 2018/2019 5th Exercise Sheet ( return Thursday November 22, during lecture ) Name(s):

Group:

5.1 Vortex motion (2 Points) Consider two horizontally oriented flow fields given in cylindrical coordinates ~va = (vr , vϕ , vz ) with vr = 0, and vz = 0, and: • Type A: A rigid body rotation“ with constant angular velocity ω0 (in the vertical ” direction), hence ~vA = (0, ω0 r, 0), and • Type B: a so called free point vortex flow where the absolute value of the azimuthal (tangential) velocity vϕ is proportional to 1/r, hence ~vB = (0, α/r, 0) (for r > 0 ) ~ × ~v in cylindrical coordinates) of the velocity fields ~vA a) Calculate the vorticity (i.e ∇ and ~vB . b) Now consider a combined velocity field with an inner part of type A and an outer part of type B, hence ( ω0 r : r R r Calculate α such, that ~v is continuous (not continuously differentiable) at r = R and sketch the velocity field as well as the value of the vorticity as a function of r . Sketch the deformation of fluid parcels in the inner and outer part after a short time of motion. c) Calculate the circulation Γ=

I

S

~v d~s

Stokes

=

Z

AS

~ × ~v) dA ~ (∇

(2)

where S is a closed path and AS is a surface surrounded by S, for a circular path at r1 < R and for another circular path at r2 > R. Comment on your result. d) Now consider a somewhat smoother velocity field with r 2 2 vϕ = U0 e−r /R R Calculate the vorticity and comment on the orientation of the vorticity vector.

(3)

5.2 Geostrophic Flow in the Ocean (1 Point) The English Channel is located at about 51◦ N and has a width of about 35 km at Dover. Due to wind-induced and tide-induced turbulence, the water mass is vertically well mixed (constant density ρ = 1025 kg m−3 ). The mean geostrophic current is directed to the North-East with a mean velocity of v = 0.2 m s−1 . a) Sketch the the isobars, the pressure gradient force, and the Coriolis force into the figure.





~  b) Calculate the magnitude of the pressure gradient ∇p  in the English Channel.

c) Compute the pressure difference ∆p between Calais and Dover.

d) Determine the difference in the ocean level ∆h between France and Great Britain. Where is the water level higher?...