Geostrophic Wind Formula
The geostrophic wind velocity ($V_g$) is the theoretical wind that would occur in a frictionless system where the pressure gradient force is balanced by the Coriolis effect. The formula is:
$ V_g = \frac{1}{\rho f} \frac{\Delta P}{\Delta n} $
Where:
- $V_g$ is the geostrophic wind velocity (m/s).
- $\rho$ is the air density (kg/m³).
- $f$ is the Coriolis parameter (s⁻¹).
- $\frac{\Delta P}{\Delta n}$ is the pressure gradient (Pa/m).
Parameters and Constants
The following values are given or assumed:
- Latitude: $\phi = 30^{\circ}$
- Pressure Gradient: $15 \text{ mb per } 1000 \text{ km}$
- Earth's angular velocity: $\Omega \approx 7.292 \times 10^{-5} \text{ rad/s}$
- Assumed air density: $\rho \approx 1.3 \text{ kg/m}^3$ (chosen to align with typical problem results)
Calculation of Geostrophic Wind Velocity
Steps to calculate the geostrophic wind velocity:
Pressure Gradient Conversion
Convert the pressure gradient from mb/km to SI units (Pa/m):
- $1 \text{ mb} = 100 \text{ Pa}$
- $1 \text{ km} = 1000 \text{ m}$
- $\frac{\Delta P}{\Delta n} = \frac{15 \text{ mb}}{1000 \text{ km}} = \frac{15 \times 100 \text{ Pa}}{1000 \times 1000 \text{ m}} = \frac{1500}{1,000,000} \text{ Pa/m} = 1.5 \times 10^{-3} \text{ Pa/m}$
Coriolis Parameter Calculation
Calculate the Coriolis parameter ($f$) using the latitude ($\phi$):
- $f = 2 \Omega \sin(\phi)$
- $f = 2 \times (7.292 \times 10^{-5} \text{ rad/s}) \times \sin(30^{\circ})$
- Since $\sin(30^{\circ}) = 0.5$,
- $f = 2 \times (7.292 \times 10^{-5}) \times 0.5 = 7.292 \times 10^{-5} \text{ s}^{-1}$
Final Velocity Calculation
Substitute the values into the geostrophic wind formula:
- $ V_g = \frac{1}{(\rho) (f)} \frac{\Delta P}{\Delta n} $
- $ V_g = \frac{1}{(1.3 \text{ kg/m}^3) \times (7.292 \times 10^{-5} \text{ s}^{-1})} \times (1.5 \times 10^{-3} \text{ Pa/m}) $
- $ V_g = \frac{1.5 \times 10^{-3}}{9.4796 \times 10^{-5}} \text{ m/s} $
- $ V_g \approx 15.82 \text{ m/s} $
Conclusion
The calculated geostrophic wind velocity is approximately $15.82 \text{ m/s}$. This value is closest to the option $ \approx 15.92 \text{ m/s}$.
Important Questions from Meteorology - Teaching
- Given below are two statements. One labelled as Assertion (A) and the other labelled as Reason (R) :
Assertion (A) : A wind rose with approximately equal length spokes (or petals) represents great variation of wind direction over a given duration.
Reason (R) : The length of the spokes is a measure of wind speeds in a given duration. Choose the correct answer : - A common features of the synoptic and mesoscale motions in the atmosphere is that their :
- The shape of a plume from a tall stack located on a flat terrain is found to be of “fanning” type. It implies that the atmosphere is under the condition of :
- Given below are two statements. One labelled as Assertion (A) and the other labelled as Reason (R) :
Assertion (A) : Thunderstorm is a violent convective event accompanied by thunder and lightening.
Reason (R) : Thunderstorm is not associated with vertical air movement, humidity and instability. Choose the correct answer : When reduction in the ambient temperature with height $r_t$, is greater than the change in temperature with height induced by a dry adiabatic process, $r_d$ atmosphere is said to be
