MATH3261/5285 ASSIGNMENT 2

• Due:  Noon Monday 24th October 2022 on Moodle

• Submit with signed assignment cover sheet

• Collaboration is encouraged but acknowledge your contributors!

1. Tornadoes and water flowing down a plughole are large-Rossby number flows, so the Coriolis force is negligible.  Instead, these flows satisfy cyclostrophic  balance, in which the radial pressure gradient force fpres = −ρ0(−)1 dp/dr is balanced by the centrifugal force fcent  = Ω2r, where ρ0  is the density of the fluid, p(r) is the pressure, Ω is the angular velocity of the rotating flow  (not that of the planet), and r is the distance from the axis of rotation.

(a) Show that cyclostrophic flow will only occur around regions of low pressure. What does this imply about the direction of rotation of the flow?

(b) Jupiter’s Great Red Spot is a 20,000km diameter anticyclone located at 22 ◦  S. Typical velocities observed in this region are of the order 50 m/s, and Jupiter rotates once every 10 hours. Calculate the Rossby number associated with Jupiter’s Great Red Spot and argue whether it is better modelled using geostrophic balance or cyclostrophic balance.

2. In this question you will derive a simple model of the atmospheric jet stream.

(a) In class, we saw that a flow that is in both geostrophic balance and hydrostatic balance is said to be in thermal wind balance.  For a Boussinesq fluid on an f-plane in the Northern Hemisphere (i.e. f = f0  > 0), thermal wind balance implies

∂u      ∂ρ          ∂v          ∂ρ

∂z     ∂y ,        ∂z        ∂x .

where for simplicity we have taken ρ0  = f0  = g = 1.  Assume that the atmosphere has a linear poleward density gradient ρ = y .  Sketch or plot lines of constant pressure p and geostrophic velocity u on the y − z plane. Show that the u increases uniformly with height but does not depend on y .  (You may assume that the surface pressure p(0) is independent of y and that the surface wind u(0) = 0.)

(b) A more realistic model of the atmosphere has a sharp density gradient in the mid- latitudes.  Assume now that the atmosphere has a density gradient that is given by ρ = tanh(y − 1), so that the density changes most rapidly at the point y = 1. Sketch or plot lines of constant pressure p(y,z) and geostrophic velocity u(y,z). At what value of y is the geostrophic velocity maximum?

3. A cold-core ocean eddy located at 40◦  S has been investigated by satellite measure- ments of sea-surface height and ship-based observations of temperature, salinity, and density.

(a) Start with the equation for hydrostatic balance

dp

Integrate this equation from depth z  < 0 to the sea surface z  = η(x,y), taking the atmospheric pressure to be p0  = constant and the pressure at depth z to be p(x,y,z). Using the resulting expression, show that the geostrophic current at the surface is

usurf(x,y) = −  g  ∂η

vsurf(x,y) =  g  ∂η

(You can take ρ0 , f0 as constants.) Find an expression for the geostrophic streamfunction ψ(x,y) in terms of the sea-surface height η(x,y).

(b) Satellite measurements of sea-surface height are fit to the function

η(x,y) = −η0 e −r2 /L2 ,

r = ^x2 + y2

where η is the surface height perturbation, r is the distance from the eddy centre, and η0  and L are constants. Calculate the surface geostrophic currents usurf , vsurf .

(c) Ship measurements of T and S were used to compute density from the equation of state. For z > −H the results were fitted to the radially symmetric function

ρ = ρ0 − az + b e −r2 /L2  ez/H ,

where z is the usual vertical coordinate (increasing upwards with value zero at the mean sea level height) and ρ0 , a, b, and H are constants.

By integrating the thermal wind equation,

∂u       g   ∂ρ

=

∂z      f0 ρ0 ∂y ,

∂v           g   ∂ρ

∂z        f0 ρ0 ∂x .

find the geostrophic currents ug(x,y,z) and vg(x,y,z).   [Hint:  find the constants of integration using the constraints u(x,y, 0) = usurf(x,y) and v(x,y, 0) = vsurf(x,y) at z = 0.]

Explain why the geostrophic current field does not depend on the variable a.

(d) The constants in the above equations are found to have the following values: η0  = 0.225 m, ρ0  = 1025 kg/m3 , L = 100 km, H = 1200 m, a = 4 × 10−3  kg/m4 , b = 0.3 kg/m3 .  Using your favourite plotting program (Matlab, Maple, Python etc), make a contour plot of the density anomaly

σ = ρ − ρ0 ,

in the y − z plane, i.e., x = 0, y in the horizontal and z > −H in the vertical.

(e) Make a corresponding contour plot of the geostrophic current field ug(0,y,z) in the y − z plane. In what direction is the eddy rotating?

4.  MATH5285 students / MATH3261 students extra credit

A Lockheed WP-3D Orion aircraft flying through a cyclone at 45 ◦ N makes measurements of windspeed and finds the following functional form for the tangential velocity uθ(r) as a function of distance r from the cyclone’s centre,

  Ur

'(')  Ua

'

(   r  ,

r ≤ a

r ≥ a,

where U is the maximum tangential windspeed located at r = a.  The cyclone is as- sumed to be circularly symmetric and uθ  is taken to be positive in the anti-clockwise direction.

(a) In polar coordinates (r,θ), the horizontal gradient operator acting on the pressure p is

∇zp =  rˆ +   θˆ,

where rˆ and θˆ are the unit vectors in the radial direction and tangential (anticlockwise) direction, respectively.  Assume that the air density ρ0  and Coriolis parameter f0  are constant. Show that the geostrophic balance relation becomes

1  dp                          dp

that is, p = p(r) is a function of r only and contours of constant pressure (isobars) form concentric circles around the centre of the cyclone.

(b) Considering separately the inner core (r ≤ a) and outer ring (r ≥ a) of the cyclone, integrate the geostrophic balance relation to find an expression for the pressure p(r). Show that, for a cyclonic system in the northern hemisphere (U , f0  both positive), the pressure has a minimum at the centre of the cyclone.

(c) In polar coordinates, the vorticity of a 2D velocity field u = ur(r,θ) rˆ+ uθ(r,θ) θˆ is given by

∇×u = ω(r,θ) zˆ,       with       ω(r,θ) =  [  (r uθ) −  ]

Using the circularly symmetrical tangential velocity ur  = 0, uθ  = uθ(r) show that

  ω0 ,

ω(r) =         (  0,

r < a

r > a,

where ω0 = 2U/a is the vorticity in the core of the cyclone. [Note that the vorticity has a discontinuity at r = a.]

(d) Using values of U  =  100 m/s, a = 400 km, ρ0   =  1 kg/m3 , and a value of f0 appropriate for 45◦ N, plot the pressure, tangential velocity, and vorticity as functions of r from r = 0 to r = 2a.