Deriving the expression for wind velocity around low pressure area

[bksree]

Hi
Can somebody tell me the steps and assumptions made in deriving the equation to calculate the wind velocity around a low pressure area ? (see the attachment fro Kleppner & Kolenkow).
I tried to derive like this :
Coordinate system located on Earth's surface with i direction towards E, j direction towards N and k direction upwards.
Initial wind velocity V = V_x i + V_y j
Ω = Ω sin λ j + Ω cos λ k where λ is the latitude of the place considered.
Position vector of parcel of air r = r_x i + r_yj

Eqn for the rotating system is
ma_rot = ma_inertial - 2m Ω * V - Ω * (Ω * r) ----- (1)

ma_inertial = -(ΔP)S j where S = cross sectional area of parcel of air

As the flow is assumed to be steady, LHS of equation 1 i.e. ma_rot = 0

RHS evaluated vectorially.

However, I cannot get the expression shown in the attachment.

Please help out. Is there any other text which shows this derivation ?

TIA

 

[eljose79]

Hello there,

Thank you for your question. The equation for calculating wind velocity around a low pressure area is derived from the principles of fluid dynamics and the Coriolis force. The steps and assumptions made in this derivation are as follows:

Step 1: Assumption of steady-state flow
In fluid dynamics, steady-state flow refers to the condition where the flow parameters (such as velocity, pressure, and density) do not change with time. This assumption is made in the derivation of the wind velocity equation as it simplifies the equations and makes it easier to solve.

Step 2: Assumption of an inviscid fluid
In the derivation, it is assumed that the fluid (air) is inviscid, meaning that there is no friction or viscosity in the flow. This assumption is also made to simplify the equations and make them solvable.

Step 3: Assumption of a rotating reference frame
To account for the rotation of the Earth, a rotating reference frame is used. This frame is attached to the Earth and rotates with the same angular velocity as the Earth. This assumption introduces the Coriolis force, which is responsible for the deflection of winds in the Northern and Southern hemispheres.

Step 4: Application of Newton's laws
The equation is derived using Newton's second law, which states that the sum of forces acting on a body is equal to its mass times its acceleration. In this case, the forces acting on a parcel of air are the inertial force, the pressure gradient force, and the Coriolis force.

Step 5: Vector analysis
The equations are solved using vector analysis to account for the direction and magnitude of the forces involved.

The derivation you have attempted is correct, but it may be missing some steps or assumptions. I would suggest referring to other texts on fluid dynamics or meteorology for a more detailed derivation. Some good resources to consider are "Atmospheric Science: An Introductory Survey" by John M. Wallace and Peter V. Hobbs, and "An Introduction to Dynamic Meteorology" by James R. Holton.

I hope this helps. Let me know if you have any further questions.