# Planetary Boundary layer
## Intro
Boundary layer is the lowest part in atmosphere, which is only a small portion. But, since the BL is near to the interface of of atmosphere and land (topography), the effect of viscosity and turbulence will be much significant than in free atmosphere.

## Non-slip boundary condition
A common-used boundary condition in atmospheric BL, there are sone properties used in this assumption:
1. Fluid has no relative motion respect to the boundary.
2. Vertical wind shear will like the figure below:

The horizontal wind obtaining a vertical wind shear within BL, and which at free atmosphere is governed by weather system.

## Turbulence
### Properties
1. quasi-random motion
2. In boundary layer, the turbulence is regarded as small scale.

### Source
1. Thermally; statically unstable
2. Dynamically: vertical wind shear

### Quantitative index of environmental condition for turbulence -- Richardson number ($R_i$)

1. Definition:
$$
R_i = \frac{N^2}{|\frac{\partial \vec{v}}{\partial z}|^2}
$$

With the bouyancy frequency is greater, $R_i$ will be greater. And when vertcial wind shear is greater, $R_i$ will be smaller

2. Criteria
When $R_i < 0.25$, the atmosphere is regarded as unstable under this perspective (this will be proven in AD II). When the atmosphere is unstable, Kelvin-Helmholtz instability will release and form turbulence.

When $0<R_i<0.25$, the atmosphere is unstable, and the source is dynamical source.
And when $R_i<0$, the atmospehere is unstable casued by thermal condition.

3. Examples
* Daytime, nighttime
in daytime, the stability is small and vertical wind shear is greater than nighttime, so the turbulence is easy to occur.
* Convection
Within convection cluster, the $N^2$ is usually negative, so it is common to see turbulence.

## Raynold average
### Basic Property
1. Since the turbulence is small in both temporal and spatial, we usually will not consider it when discussing large scale or mean flow question no matter in free atmosphere or boundary layer.
2. Definition
For some property $a$, it can be written as:
$$
a = \overline{a} + a^\prime
$$

where:
$\overline{a}$ is defined as:
$$
\overline{a} := \frac{1}{\Delta t}\int_0^{\Delta t} a\;dt
$$

,which is the time mean of a property to vanish small fluctuation.

3. Spatial mean
Since the property can be regarded as mean flow and fluctuation, its spatical averaging can be written as:
$$
\om $$

###