# AP3001

#### **(10 points)** Derive the thermal wind equation $\frac{\partial V_g}{\partial \ln p} = \frac{-R}{f}k \times \nabla _p T$

* 2011 3(a)
* 2019 2(a)
* 2016 3(a)

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#### **(20 points)** Gradient wind equaiton 梯度風 $\frac{V^2}{R} + fV = -\frac{\partial \Phi}{\partial n} = f V_g$

* 2019 2
* 2016 2
* 2011 2

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#### **(5 points $\times$ 2)** A Rankine vortex counterclockwise rotating at constant angular velocity

#### **(10 points)** The Circulation Theorem

* 封閉路徑環流，沿各切線的線積分

$C = \oint \vec{U} \cdot d \vec{l}$

### Past Exam

### Catogrize by the Year

* ### 2021

1. Given the gradient wind equation as $\frac{V^2}{R} + fV = -\frac{\partial \Phi}{\partial n} = f V_g$

a. Explain why cyclostrophic wind can be clockwise or counter-clockwise in **both Hemispheres**. (5%)


b. Based on the gradient wind equation, explain why the geostrophic wind is an overestimate (underestimate) of the balanced wind in a normal low (high) system. (5%)


c. Explain why a solution exists for an abnormal low in **Southern Hemisphere** and draw the flowsystem (wind and pressure) and the balanced forcings for the solution. (5%)


d. Explain why the $\frac{\partial \Phi}{\partial n}$is more limited for high pressure systems than low pressure systems. (5%)

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2. ~

a. Derive the thermal-wind equation in isobaric coordinates as $\frac{\partial V_g}{\partial \ln p} = \frac{-R}{f}k \times \nabla _p T$ (10%)



b. Use (a.) to explain why a westerly jet must occur near the tropopause in mid-latitude in the winter of **Northern Hemisphere**. (5%)


c. Use (a.) to show how the geostrophic wind will turn with height in Northern Hemisphere when the layer is associated with a warm air advection from the equator. (5%)


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4. A Rankine vortex counterclockwise rotating at constant angular velocity $\vec{\Omega}$ upward $|\vec{\Omega}| = \Omega$ indeed has purely tangential velocity given by $\vec{V} = \vec{\Omega} \times \vec{r}$

a. Determine the vertical vorticity at the circle with a radius of R centered at $r = 0$. (5%)


b. Determine the circulation of any square with each side of R centered at $r = 0$. (5%)


**{ Hint: use Stokes' circulation theorem F }**

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5. Derive vertical vorticity consisting of two components, (shear vorticity and curvature vorticity) in natural coordinates and explain why an intense cyclonic typhoon may not have cyclonic vorticity at some distance outside of the radius of the maximum wind. (10%)

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* ### 2019

3. Show that the absolute circulation is conserved following the air motion for a closed path on an isentropic surface. (10%)


5. Vertical vorticity consists of two components (shear vorticity and curvature vorticity) in natural coordinates (don't need to derive). Explain these two components and show how an intense cyclonic typhoon may be associated with anti-cyclonic vorticity outside of the radius of the maximum wind. (10%)