# Atmospheric Dynamic II - Quiz 1

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### (1102-1) Must!!!
For the two laver model. its instability regime is given by the right diagram and the phase speed is given by $c = U_m - \frac{\beta(k^2 + \lambda^2)}{k^2(k^2 + 2\lambda^2)} \pm \delta^{1/2}$ where $\delta \equiv \frac{\beta^2\lambda^4}{k^4{(k^2 + 2\lambda^2)}^2} - \frac{{U_T}^2 (2\lambda^2 - k^2)}{(k^2 + 2\lambda^2)}; \lambda^2 \equiv \frac{{f_0}^2}{[\sigma(\delta\rho)]}$

Discuss:
* [ ] the setup of the two-layer model,
* [x] stability of short waves. 
* [x] the magnitude of minimum $U_T$ for instahility to occur. 
* [x] the role of $\beta - effect$ 
* [x] the most unstable (optmal) for instability(from the diagram) and
* [x] the growth rate for the condition of $\beta = 0$



### (1102-2)
* [x] Diagram the structure of most amplifying baroclinic wave at upper and lower levels in the two-layer model, and discuss the roles of perturbation vertical circulation $(\bar{w^\prime T^\prime})$ and meridional circulation $(\bar{v^\prime T^\prime})$ in energy conversion for mean flow and eddy.

### (1102-3) Available Potential Energy (APE)
* [x] Show that for a hvdrostatic system $E_p=\int_{0}^{\infty}\rho gz dz = (R/C_v)E_1$
* [x] Discuss available potential energy (APE) and give one simple example to cxplain APE.

### (1102-4) Rayleigh
The Rayleigh Theorem indicates that the quasi-geostrophic (O-G) system satisfies the constraint where $z^*$ is the log-pressure height with the flow top H.
* [x] Please explain how both barotropic instability and baroclinic instability exist in the system.
* [ ] Explain how barotropic instability is related to the meridional gradient of absolute vorticity.

### (1102-5)
Brietly answer the following problems
* [x] Derive the mean incompressible continuity equation using Reynolds average and decomposition.
* [x] What is TKE? Describe how the TKE can be produced or lost in the TKE equation.
* [x] Show that vertical turbulent momentum fluxes are always downward near the lowest surface.
* [ ] Express turbulent momentum and heat fluxes based on K-theory.
* [x] Explain why the Navior-Stokes equations without Reynolds average are not applicable in practice.

### (1102-6)
* [ ] Express the Reynolds-average large-scale momentum equations with K-theory and discuss how the steady-state mean flow in balance will be crossing isobars toward the lower pressure in PBL.
* [ ] Depict the Ekman spiral of the wind solution for constant $K_m$ and estimate the depth of the Ekman layer with $K_m = 5 m^2s^{-2}$.

### (1102-7) The surface layer
* [ ] Explain why the surface laver is usually called as a constant-flux layer and use this assumption to derive the wind profile in the neutral surface layer based on K-theory.

### (1102-8) Secondary Circulations & Spin Down
* [ ] Qualitatively discuss how Ekman pumping can occur due to the presence of turbulent friction in the planetary boundary layer, and how such a pumping may cause the primary baroclinic cyclone to spin down effectively through the induced secondary circulation.

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TODOs:

* [ ] Re-organize the question with their own topic below.

## Rayleigh