## Problems

1. Define a streamfunction y for nondivergent, two-dimensional flow in a vertical plane:

du dv

dx dy and interpret it physically.

Show that the instantaneous particle paths (streamlines) are defined by y = const, and hence in steady flow the contours y = const are particle trajectories. When are trajectories and streamlines not coincident?

2. What is the pressure gradient required to maintain a geostrophic wind at a speed of v = 10 m s-1 at 45° N? In the absence of a pressure gradient, show that air parcels flow around circles in an anticyclonic sense of radius v/f.

3. Draw schematic diagrams showing the flow, and the corresponding balance of forces, around centers of low and high

TABLE 7.1. Summary of key equations. Note that (x, y, p) is not a right-handed coordinate system. So although z is a unit vector pointing toward increasing z, and therefore upward, Zp is a unit vector pointing toward decreasing p, and therefore also upward.

(incompressible—OCEAN)

(comp. perfect gas—ATMOS)

Continuity dp dt

dp gP

Geostrophic balance fu = p z x Vp fu = phJ x vp rref fu = gZp x Vpz

Thermal wind balance (2Q ■ V)u = p Vp x p Vp f du = - pL z x Vff

pressure in the midlatitude southern hemisphere. Do this for:

(a) the geostrophic flow (neglecting friction), and

(b) the subgeostrophic flow in the near-surface boundary layer.

4. Consider a low pressure system centered on 45° S, whose sea level pressure field is described by

p = 1000hPa - Ape where r is the radial distance from the center. Determine the structure of the geostrophic wind around this system; find the maximum geostrophic wind, and the radius at which it is located, if Ap = 20 hPa, and R = 500 km. [Assume constant Coriolis parameter, appropriate to latitude 45° S, across the system.]

5. Write down an equation for the balance of radial forces on a parcel of fluid moving along a horizontal circular path of radius r at constant speed v (taken positive if the flow is in the same sense of rotation as the Earth).

Solve for v as a function of r and the radial pressure gradient, and hence show that:

(a) if v> 0, the wind speed is less than its geostrophic value,

(b) if |v| <<fr, then the flow approaches its geostrophic value, and

(c) there is a limiting pressure gradient for the balanced motion when V > -1/2fr.

Comment on the asymmetry between clockwise and anticlockwise vortices.

6. (i) A typical hurricane at, say, 30° latitude may have low-level winds of 50ms-1at a radius of 50 km from its center: do you expect this flow to be geostrophic?

(ii) Two weather stations near 45° N are 400 km apart, one exactly to the northeast of the other. At both locations, the 500- mbar wind is exactly southerly at 30 ms-1. At the northeastern station, the height of the 500-mbar surface is 5510 m; what is the height of this surface at the other station?

What vertical displacement would produce the same pressure difference between the two stations? Comment on your answer. You may take PS = 1.2 kgm-3.

7. Write down an expression for the centrifugal acceleration of a ring of air moving uniformly along a line of latitude with speed u relative to the Earth, which itself is rotating with angular speed Q. Interpret the terms in the expression physically.

By hypothesizing that the relative centrifugal acceleration resolved parallel to the Earth's surface is balanced by a meridional pressure gradient, deduce the geostrophic relationship:

1 dp pdy

(in our usual notation and where dy = ady).

If the gas is perfect and in hydrostatic equilibrium, derive the thermal wind equation.

The vertical average (with respect to log pressure) of atmospheric temperature below the 200-mbar pressure surface is about 265 K at the equator and 235 K at the winter pole. Calculate the equator-to-winter-pole height difference on the 200-mbar pressure surface, assuming surface pressure is 1000 mbar everywhere. Assuming that