Fluids Dynamics and Quasi-Geostrophic Equations - Math Problem Example

20* 1000= 2000

10/100*2000 = 200

200/60 = 3.3

So the answer is 3.3m/s

(ii)

dV/dt = −V 2/LF

LF = 3m/s * 1 hour and L = 50 km

50 * 1000 = 50,000.

So

dV/dt = 50, 000/3 = 16, 666.7m/s

(iii)

Dθ / Dt = kH ∂2θ/ ∂x2

Estimate of a

16, 666.7m/s = kH ∂2θ/ ∂x2

So

16, 666.7 = kHz * 580

Hence kH = 16,666.7/580 = 28.

This is how the extent of the real sea breeze is determined.

No 3

Quasi-Geostrophic (QG) equations

At the momentum level and making use of the pseudo height coordinate z = [1 − (p/p0)κ]cpθ0/g the equations are

They are used in large-scale atmospheric flows in addition to some oceanic flows. However, at times the Rossy number is small, and hence at times the inertial accelerations can be neglected (Masaki 2004).

Planetary geostrophic equations

Planetary geostrophic equations are a standard model of thermohaline circulation. They are obtained from equations such as the Froude number Fr, the Rossby number, as well as the Burger number Bu going to 0.

These equations are of tremendous diagnostic value. A pressure field allied to geostrophic velocities is an applicable solution as long as boundary requirements have been satisfied. However, these equations do not give any information regarding how the flow evolves (Chelton 2006).

B

(i)

Where the potential vorticity q is

(Chelton 2006).

(ii)

At first, we begin with disturbing a flow that with only a time and spatially invariant zonal flow U without a meridional component.

We assume the perturbation to be a lot smaller than the mean zonal flow.

Supposing non-divergent flow which stream function entirely describes the flow,

Which, and plug to get: 4

A traveling wave solution having wave numbers k and l, in addition to frequency omega:

5

The dispersion relation of the below equation is obtained:

 6

Both zonal phase speed together with group speed is given by the following equations respectively:

(Chelton 2006).

(iii)

• c is the phase speed,
• CG is the group speed,
• u is the mean westerly flow,
• β is the Rossby parameter,
• k is the zonal wave number.

The above indicates that phase speed is at all times westward relative to mean flow. The meaning of large and small is entirely dependent on the value of l, if l = k, subsequently, the group speed is equivalent to the mean zonal flow.

Vertical in-uniformity within the stratification is given by:

And the β plane approximation is used to take in the end product of the variation in f with latitude,

The dissipation operator shows the effects of every scale of motion smaller than those unambiguously resolved in the numerical calculation; characteristically this equation is used which is hyper-viscous diffusion:

, where is a small hyperviscosity.

Therefore there are only two major variables in the QG code: and q.

(Chelton 2006).