EART60071 M&P 2

 Measuring and Practice 2 Semester 1

Week1 (7): Tools of the trade (学习工具): 

Week 2 将会使用使用Parcal Model来模拟properties of air in a parcel, 包含一些水蒸气气雾粒子的空气太阳光照射下发生的状况。

Week3 将会使用empirical Model来解决在不同情况下大气中空气污染物移动，温度不同导致情况不同

Week4 使用Reduce Dimension Model & Single Column Model, 研究丘陵和山地对于降水的影响

Week5 使用Shallow Fluid Model，可以研究流动模型的性质，对于洋流或者大气都有用。可以对于气候和气象都可以用。

Computer practice: 

使用的代码放在了另一博文中。本实验主要进行求解一套微分方程的近似解。用一个放射性粒子衰变方程作为例子。

使用前向欧拉法（正向欧拉法）进行近似求解：前向某点的切线斜率与dt的值

P1增加dt的值之后呈现三种情况：

在这个实验中dt的变化其实并没有影响到解析解的值，因为首先解析解是通过公式严格推导得到的一个准确的解，所以解析解再这里是时间的函数，不管评估函数的时间间隔如何，它的值不会变化。但是欧拉解的准确性降低。因为时间间隔变化梯度区间变化了，dt越大越偏离真实解。

P2二阶常微分方程

使用二阶常微分方程研究dt增加的情况：

dtsolve=0.5

dtsolve=0.001

在这里减少近似解的时间步长就可以增加准确度，当曲线随时间快速变化的时候解的准确度下降。

P3偏微分方程

研究传输方程沿x轴以常数传输三角函数，这是纯粹的传输：

虚线为解析解，当dt=6的时候由于时间步进太长了，所以导致近似解完全不稳定，数值爆炸。

Week2 (8): Box and Parcel Models in Environmental Science

Lecture: Box and Parcel Models in Environmental Science

1. Parcel Models in Environmental Science: First indirect effect of aerosols on clouds. 气溶胶对云的第一次间接影响。

主要的应用有: Geoengineering Scheme: Marine Cloud Brightening

2. 气溶胶颗粒的影响：

Kelvin Effect：开尔文效应是考虑温度和压力对粒子运动影响的结果，将粒子换成水蒸气：

Droplet growth from water vapour：水蒸气里面的水汽粒子在一定的压力和温度下增长。空气膨胀中冷却的时候水蒸气变成气溶胶颗粒water vapour eventually condenses onto aerosol particles known as Cloud Condensation Nuclei 称之为云凝核。

3. 在亲水和疏水的毛细管中的毛细作用

由于弯曲的企业表面导致的蒸汽压变化，可以通过开尔文方程来计算。For liquid water to condense onto aerosol particles the actual water vapour pressure must be higher than the equilibrium vapour pressure. 液态水冷凝到气溶胶颗粒上必须要实际蒸汽压大于平衡蒸汽压。

Bin / Sectional assumption： 将不同的气溶胶颗粒通过尺寸与质量分类，解决中心增长问题，这样的一组就被称之为 Bin 或者 sectional.

更多的serosols能够反射更多的阳光，虽然水蒸气被均分的更细了，但是能更多的反射阳光。就像是空中飞机飞过的时候的线就是飞机的污染颗粒形成的气溶胶。

Initialisation: aerosol 'modes' / size distributions：气溶胶颗粒的大小tend to 'clump' into  modes. 他们比较倾向近似于正态分布。他们这些气溶胶颗粒可以是盐、有机物等等，they act as CCN, sea salt aerosol添加到背景中作为更有效的CCN增加Cloud drops.

Computer practical: Parcel Modelling

In the atmosphere, aerosol partical tend to come in 3 modes: 'the nucleation mode' 'the accumulation mode' 'coarse mode' this 3 mode will set as defaut mode into the input file and water will condense onto these aerosol particles to form a cloud.

The project aiming to add sea salt aerosol particles to the background aerosol will alter the properties of the clouds making them more reflective.(小声bb：更多的云意味着更多的反射阳光，将阳光反射回太空中，似乎也是一种减缓全球变暖的途径？)

Default set:

P1: Temperature lapse rate of the parcel

默认情况：使用背景气溶胶，以0.3米每秒的速度穿越大气层上升到1200m的高度

winit=0.3m/s

The rate of the line between ground and 800 is 8 degrees, after 800 to 1000 is 10 degrees. This is known as the dry adiabatic lapse rate（干燥绝热气温垂直递减率）
超过800m的时候 the air is saturated, so any further cooling results in condensation of liquid water and the release of latent heat. 任何冷却都会导致水凝结放热。

P2: Effect of vertical wind-speed

垂直风速，空气冷却的速度对于云的形成也非常重要，现在需要改变参数 'winit' 也就是垂直层面整个parcel上升的速度从默认的0.3改为0.03和3.0，之后再次进行模拟

winit=3.0m/s

winit=0.03m/s

首先可以看到的是气温的垂直递减率(lapse rate)是没有变化的。

在地面处的初始湿度是相同的，而且干绝热垂直递减率没有变化，所以导致云层高度其实没有变化都是在800m。Cloud base height is the same, because the initial humidity is the same(ground) in each case, and because the dry adiabatic lapse rate is also equates to the same height.

When changing the vertical wind speed, the mixing ratio of top left plot is roughly the same as before(0.65), because in a given cooling condition is the same in a parcel regardless of the speed. 当改变垂直风速的时候混合比率并没有变化，因为给定的冷却条件下在一个区块的冷却凝结都是一样的。The droplet concentration increases for the faseter vertical wind speed, but decrease for the slower vertical wind speed. The faster cooling force water vapour to condense on even smaller aerosol particles, because there is a delay in the time it takes for water vapour to condense. Because of this the humidity can build up to become more super-saturated, which allows water to condense on highly curved surfaces.更快的冷却速度可以让水蒸气凝结在更小的气溶胶颗粒上，因为水蒸气凝结需要一定的延迟，因为这些延迟可以让水蒸气更加饱和，也就是说可以让水在更加弯曲的表面凝结。对于更慢的冷却情况相反。

P3 Increasing humidity(water vapour) 增加湿度

改变地面湿度rhinit=0.95

Increasing the initial humidity made the cloud form at a lower attitude, this because less cooling / ascent was needed to saturate the air. The mixing ratio at the cloud top is also higher(0.95), This because compared to the default case, the cloud formes any further cooling results in condensation. 增加了相对的湿度能够让云在更低的海拔行程，因为一旦云形成之后需要更少的冷却或者上升就能够让空气中的水饱和。所以在云顶的混合比率更高。因为一云形成之后进一步的冷却也能够造成凝结。

P4 Adding sea-salt aerosol 向空气中加入海盐凝结核

现在向空气中加入海盐凝结核，运行模拟。（默认情况为150/cc）

1. 浓度1000x10^6

2. σ8=0.01

3. 浓度100x10^6

浓度越高海盐的分布越广，也就是说更多的海盐颗粒激活成了云滴，他们的大小非常相似。The higher the concentration, the wider the distribution of sea salt, which means that more sea salt particles are activated into cloud droplets, and their sizes are very similar. 虽然大颗粒比小颗粒更容易生成，但是这些颗粒是相同大小的，所以他们一起生长。Large particles can grow easier than smaller particles, but these are all the same size, so they all grow together. 如果降低了海盐浓度(3)，云滴数也比默认情况下小。因为海盐的气溶胶颗粒大，较大的颗粒比较小的颗粒生长更快，并且现在海盐浓度更低，所以导致海盐更能争夺来水分，背景气溶胶失败。If the sea salt concentration (3) is reduced, the number of cloud droplets is also smaller compared to the default case. This is because sea salt aerosol particles are larger, and larger particles grow faster than smaller ones. With the sea salt concentration now lower, it results in a greater ability for sea salt particles to compete for water, causing the background aerosol to lose out.

Week3(9): Transport and Mixing of Pollutants: Gaussian Plume Model 污染物的传输和混合：高斯羽流模型

Lecture: Gaussian Plume Modelling

1. Lagrangian approach vs the Eulerian approach拉格朗日方法和欧拉方法

拉格朗日方法：用来解决solar system model and the parcel model, 可以适用1d~3d的模型并且xyz可以取任何值，并且模拟在坐标下定义的变量。

欧拉方法：可以是1d~3d，在xyz空间中固定的位置并模拟这些变量。

2. The importance of the temperature lapse rate 气温递减率的重要性

在对流层中干燥的空气每上升一公里下降10摄氏度 In the troposphere rising dry air cools at 10 degC per kilometer. 湿空气是6摄氏度，空气本身可能就有不同的递减率。

Above tropopause, lapse rate is governed by other processes, such as the break down of ozone by UV. 在对流层上方空气递减率可能由于其他的过程影响例如臭氧被紫外线分解。In the roposphere, lapse rate is very important because is governs the vertical stability. 在对流层中递减率非常重要，它决定了垂直稳定性。

左数图一称之为Stabel air: 带有点的实线代表了正常的大气环境，也就是measure the atmosphere this is the temperature lapse rate that we would measure，虚线代表了一小团空气可能走过的路径。It cools at this known rate of 10 degrees, if a parcel cools faster than environment, then it ends up colder than the environment,this air will be more sink, 热空气向上走，冷空气向下走，这样叫stable air 因为它们更倾向于各就各位(if the air moves upwards from where it is, the tendency is for it to sink back down where it started from)

中间图称之为Neutral environment中性环境，if the air rises, there's nothing to stop the air from rising, but the air does need something to force the air up or down. 我们需要一些东西去推动air运动，否则这些空气则会停留在原处。

最右边的图parcel air cools at a rate that's slower than the atmosphere, the tendency is that the parcel will want to rise.它最终会比大气层更热，所以倾向于向上升。

空气越不稳定，扩散和混合就越多，并且我们需要考虑大气扩散和颗粒物行为，这样就牵扯到了空气湿度，因为空气湿度能够影响颗粒物的大小。烟囱扩散距离，位置，风口，横风位置以及烟囱的排放量都决定了地面的污染程度。The more unstable the air is, the more dispersion and mixing there is. And also we often need to take both atmospheric dispersion and the behaviour of the particulate, this will be interupted by relative humidity, the humidity changes the size of these particulates. The distance from the stack, the positioning of where you are These factors all determine the pollution situation。

Q:

1. Gaussian curve is also called a bell-shaped curve.

2. Two approaches to numerical models are: the Lagrangian approach and the Eulerian approach.

3. A rate at which the air temperature decreases with height in the atmosphere is referred to as lapse rate.

3. Stacks烟囱:

Geometry几何学：进行模拟的动机主要是工业烟囱收到排放监管，需要获得排放许可并且遵守相关法规。The main motivation for conducting the simulation is that industrial chimneys are subject to emission regulations and require emission permits and compliance with relevant regulations.

Assumptions 假设：

左边从上到下不稳定中性和稳定

高斯烟囱模型主要用于模拟不同大气稳定性条件下的情况：在不稳定情况下由于包裹上升更慢所以导致温度更高，热气团上升的过程中与污染烟团相互总用形成回旋 In this unstable case, this parcel warmer than the envrionment so the parcel will interact with the pollution plume when the parcel rising，中性条件污染物还是会与空气进行混合，但是混合程度不及不稳定状态，所以倾向于出现圆锥形状称为coning In the neutral condition, the environment has a similar lapse rate to that of a parcel, and in this case pollution from the stack can still mix into the environment, but it doesn't mix as fast as in the looping case. It tends to mix out in a cone-like shape, call this coning，稳定情况下包裹上升变冷所以污染物能够控制在稳定的一个薄层中在水平层上扩散开所以这种称之为扇形 In really stable case, the parcel rises and colder than its surroundings. The pollution remain a very thin layer in the vertical but it can still spread out in the horizontal, this is referred to as fanning. 出现这些情况的主要是由于大气结构所决定的 based on the structure of the atmosphere

Analytical solution 解析：控制烟囱下风污染物的传输混合的方程称为平流-扩散方程the equation that governs the transport and the mixing of pollution downwind of this stack is known as the advection-diffusion equation.

simulate 模拟：我们可以模拟时间相关的污染物分布是曲线，但是现在忽略时间依赖性只使用时间平均条件，这样在垂直层面就呈现高斯分布。we can simulate the pollution emitted, if it is time dependent, it meanders, but we neglect the time dependence, so we just have the time averaged condition which gives a Gaussian in the vertical. 垂直方向上中心线上最大两边减小，这取决于大气的稳定程度。如果在垂直面上做一个横截面图，可以得到一个高斯分布，但是在横风截面上也可以得到一个高斯分布。

Q:

1. When industrial facilities are applying for emission permit, they tend to rely on dispersion models. 扩散模型

Answer: True. This type of model is used to assess the impact of point source emission.

2. Coning is linked to the shape of pollution cloud formation. 

Answer: True. The pollution still mixes with the environment but not as fast as in the looping case.污染仍然与环境混合，但速度没有循环情况那么快。

3. The advection-diffusion equation is used to describe the transport of the pollution only.

Answer: False. The equation is used to describe both transport of any pollutant (marked as C) and mixing of the pollution.

Computer Practical: Gaussian Plume Model

主要研究的是来自点源污染物的平流和湍流扩散问题，称为高斯羽流模型 In this handout we look at the problem of advection and turbulent diffusion of material from a point source, such as industrial stack. The result is referred to as a Gaussian Plume model.

模拟用到的stable1~6分别从Very unstable 到Very stable

default: 

very unstable prevailing wind（盛行风）

P1: Wind direction 风向:

First we look at the effect that the assumptions about wind direction have on the dispersion of pollutants.

neutral condition & constant wind（持续风）

Fluctuation wind 波动风

Prevailing wind 盛行风

污染物被风吹到了烟囱的下风方向，风向变化越大，地面污染物的数值就越低。The pollution is blown downwind of the stack by the wind, the more variable the wind direction the lower the ground level concentration of pollution is.

P2: Multiple stacks

通常一个开发区有多个烟囱，在此实验多个烟囱的影响。Often a development can have multiple stacks. 风的类型选用盛行风，单烟囱情况参照P1图三

2 Stacks

3 Stacks

In general, having more stacks increases the pollution concentration at ground. There is just more pollution being emitted from multiple stacks.

P3: Vertical stability—plan view 垂直稳定性—平面图

We want to see what effect vertical stability of the atmosphere has on the ground level concentrations.大气垂直稳定性对地面浓度的影响。选用单烟囱的情况来模拟，从左到右从上到下

进行的模拟

Unstable air result in rapid mixing very close to the stack, hence ground level concentrations are high close to the stack, but low far away. 不稳定的空气导致烟囱附近快速混合，靠近烟囱的浓度高，稳定的空气导致靠近烟囱浓度低o，空气不会快速混合到地面所以原理烟囱浓度高。污染物最终会回到地面。

P4 Vertical stability- vertical structure 垂直稳定性—垂直结构

模拟垂直方向的情况：

Height slice 高度切片

As the picture shows, very unstable air distributes the pollution in the vertical very quickly, close to the stack, because looping results in rapid mixing. Neutral air pollution less quickly, coning occurs as the air spreads in a cone shape. Stable air results in very slow mixing, the air trapped in a layer, the pollution concentration decreases with distance from the stack because it spreads mostly in 2d due to fanning.不稳定的空气让污染物在垂直上迅速分布尤其是靠近烟囱的地方，因为环流导致快速混合。中性空气在垂直方向上分布慢，因为锥形扩散。稳定的空气导致混合非常缓慢，空气被困在一层中，呈现2d分布，由于扇形效应导致。

P5: Annual cycle in vertical stability

以上模拟全部假设垂直稳定性在所有时间都是恒定的，但是垂直稳定性会随着时间和气候变化，夜晚和冬季更容易出现稳定的空气。The simulations so far have assumed constant vertical stability at all times. However, vertical stability varies with time and climate, with stable air more likely during nights and winters.

接收器位于x=0, y=-2500

In the summer, the air is unstable, due to rapid mixing in the vertical, the concentration have dropped to almost 0 (we have looping) 夏天不稳定的空气由于有环流所以导致垂直快速混合所以浓度几乎归零。In the autumn, the air is approximatelyu neutral and have some mixing ( we have coning) 秋天中性空气有一部分混合，由于锥形效应，所以地面浓度可能较高。In the winter, the air is stable, this mean we have fanning and the air is trapped aloft, the air doesn't mix to the ground.冬天空气稳定所以有扇形效应，空气被困在高空所以没有办法混合到地面。

P6: Kohler equations and map over-lay 科勒方程和地图叠加

We will look at the effect of humidity on the growth of the aerosol particles and the resulting particulate matter.

默认为Dry Aerosol, sodium cholride 氯化钠气雾剂

default

Humidify

Humidify & Organic acid

Humidification increase the mass loading of the aerosol bercause the aerosol take on water under high humidity. Common salt is generally more hygroscopic than an organic acid.更潮湿的空气增加了气溶胶的质量荷载，高湿度下气溶胶吸收水分，氯化钠比有机酸更亲水。The humidification does not affect the dispersion of the aerosol, they mix the same, but it does increase the mass loading. In reality larger aerosol particles would also fall out or sediment out of the atmosphere due to their larger settling speeds湿化增加了气溶胶荷载，但是没有影响气溶胶扩散，他们有相同的混合方式。现实中较大的气溶胶颗粒能够更快的沉降出去，但是模型里面没包含。

Week4(10): Single Column Modelling: Modelling Orographic Precipitation单柱建模：地形降水建模

Lecuture: Single Column Modelling

1. Scenario we are modelling

在云层经过山脉的时候会有一部分云内的水以雨的形式落回地面。

2. Cloud and Precipitation Processes Primer 

水汽和云凝结核形成更大的云滴的过程会导致云滴的数量减小但是增大他们的大小，以至于可以逆向空气做垂直运动作为雨滴来到地面。This process reduces the number of droplets but increases their size such that the droplets can start to fall against the vertical motion of the air and reach the ground as rain. This is known as the warm rain process because it does not involve the ice phase. 如果云层在低于冰点进一步加深，可能形成冰晶。冰晶的形成需要一部分气溶胶颗粒，这些颗粒称为冰核颗粒。雨滴在下落的过程中进一步收集水滴增加体积。

If the cloud deepens further below temperatures of 0 degrees Celsius, then ice crystals may form. The formation of ice crystals requires rare aerosol particles for the ice to form on. And these are known as ice nucleating particles. 

（暖雨过程与冷雨过程）

Q:

1. In the Tenerife example, discussed in the video above,  there are more lower clouds in the north (where they form) and the clouds get higher towards the south because ...

as the clouds move over the mountain, some of the water within the clouds falls out as rain (and so there's less water in the clouds).

2. The main transport mechanism for water vapour and aerosol particles is knows as ... advection.

3. The growth of  cloud drops from water vapour is described by ... 

an ordinary differential equation, first derived by James Clerk Maxwell.

3. Lagrangian approach vs the Eulerian approach – The advection equation (a Partial Differential Equation, PDE平流方程) 

Lagrangian: If a variable is conserved in a parcel, its rate-of-change in a frame of reference that follows the parcel is zero，这个包裹是拉格朗日的，就是说这个包裹的参考框架中的变化率为0

Eulerian: We can represent a conserved variable in a frame of reference that does not move. Apply the chain rule.在一个不动的参考框架中假设一个守恒的变量，变量是时间和空间的函数。时间导数为0（完全看不懂）

在第一周的平流方程模拟（最后一个模拟）中演示了时间步长小于1，否则不稳定可能会导致数值扩散。数值扩散更多的是选择近似方法的一个产物。

4. Details of the Model 

– Solving ODEs常微分方程 and PDEs平流方程 together 

略，完全学不懂

5. ‘Conserved’ variables 

– Potential temperature 

– Water vapour 

潜在温度由θ 表示，潜在温度沿着路径是守恒的，θ 可以通过温度和压强的组合来计算，方程如下Potential temperature given the symbol θ, and θ is conserved along the path. θ can be calculated by this combination of temperature and pressure.

如果这个parcel的压力变化，就会导致温度变化，但是θ 由于是一个空气的特性，当空气被平流或者移动到不同的压力区域的时候不会变化，但是它可能会由于凝结而改变。我们可以经过空气平流然后求解新的T。If the pressure of a parcel changes, its temperature changes, but θ is a property of the air that doesn't change when the airis advected or moved into a region of different pressure but it can change due to condensation. We can advect θ with the air and then rearrange to find the new T.

6. Equations of the model

Q1: 

Question: In a Lagrangian frame, if a variable is conserved in a parcel, what is its rate of change?

Answer: It's zero.

Question: What will happen if the product of the time-step, wind-speed and reciprocal of grid spacing, chosen for the advection equation is more than 1?

Answer: The solution is unstable.

Question: In the Partial Differential Equation in the single column model, what is velocity used? 

Answer: It's air updraft minus the fall-speed of the particles.

Question: What is the name for a conserved variable used by atmospheric models?  

Answer: It's called potential temperature and is designated as theta.

Q2:

1. The importance of clouds is linked to their ability to reflect the rays of the sun.  

Answer: True. As a result, they can stop heat from escaping and lead to rain on the ground.

2. The only condition for ice particle formation is zero degree temperature.  

Answer: False. What is needed is an ice nucleating particle for the ice to form around.

3. The experiment shown in the video uses gypsum powder to imitate aerosol particles. 

Answer: True. These are present in the atmosphere in small amounts.

4. The experiment shown in the video allows us to say exactly when a drop will freeze.  

Answer: False. The process is statistical so we can only characterise the average freezing temperature.

Computer Practical: 

A single column model of the atmosphere(SCM) is uesd to demonstrate a number of phenomena(the importance of aerosol-cloud interactions; the importance of the warm-rain process; and the importance of desert dust as ice nucleating particles (INP). This model solves equations for how aerosol and cloud particles evolve into precipitation particles. These equations are used in many weather forecasting models.

使用了大气的单柱模型（SCM）来演示一些现象，包括：气溶胶 - 云相互作用的重要性；暖雨过程的重要性；以及沙尘作为冰核颗粒（INP）的重要性。这个模型解决了关于气溶胶和云粒子如何演变成降水粒子的方程。这些方程在许多天气预报模型中得到应用。

Default:

P1 Changing Cloud Droplet Number 更改云滴数量

N cloud=10 /cc (10. e6 lower than normal)

N cloud=1000 /cc (higher)

N cloud=2000 /cc (very polluted)

云在越过山脉的时候由于云滴正在发生凝聚和碰撞所以形成雨滴, The cloud disappears as it rises over the mountain due to the droplets collide and coalesce to form rain drops

在默认情况下雨在山的上坡下落，随着云滴浓度增加雨越来越多地朝下风方向下落，有可能落在下坡侧。The rain falls on the upward slope of the mountain, mainly. As we increase the cloud drop number concentration the rain falls more and more downwind, and may even fall on the “downward slope” side.

P2: Switching off the Warm Rain Process 关闭暖雨过程

如果关闭了暖雨过程，云会变得更持久，降雨量为0。这表明在默认情况下暖雨过程非常重要，当云低浓度非常高的时候情况就更明显了。在这种情况下云滴非常小，并且下降速度很慢，所以云滴不会落入云层下的干燥空气中。If the warm rain process is turned off, the clouds become more persistent, and the rainfall is zero. This indicates the significance of the warm rain process under default conditions, and the importance becomes even more pronounced when the cloud drop concentration is very high. In such situations, the cloud drops are very small, and their descent is slow, preventing them from falling into the dry air below the cloud layer.

P3: Cooler surface condition 较冷地表情况

Set the surface temperature to 280K, the cloud base temperature 270K, cloud-top temperature 268K. There will show some ice in the simlations.

N dust = default

Then set the cloud-base temp=293.15K, cloud-top temp=288.15K

N dust=1000 /cc

In the simulation without warm rain there is rainfall, because there have cold rain being produced by the action of ice nucleating particles and ice phase processes. Ice melts and falls as rain, but there is not much rain in this case.在没有暖雨的模拟中足够低的温度导致冰核颗粒和冰形成了冷雨，冰降水融化并以雨的形式落下，在这种情况下降水量并不多。

Increasing the number of dust particles also increase the number of ice crystals, Dust particles are ice nucleating particles.增加尘埃数量能够导致云中冰粒子更多因为尘埃颗粒是冰核颗粒。

P4: Deeper Cloud 更厚的云

Change the N cloud to 1000 /cc and ∆T=5 可以看到云大概1km deep, change the ∆T=2 and run the simulation again.

∆T=5

∆T=2

更深的云层导致更多的凝结在云层高处发生，这导致云滴更大，暖雨效果更有效，并且雨水同样形成在上坡处 When the cloud is deeper more condensation can take place higher up in the cloud (as shown in the parcel model simulations). This means the cloud drops will be larger and the warm rain process will be more effective, and it forms on the upward slope first for the deeper case.

Week 5(11): Cartesian Shallow Water Model: Tsunamis and Large-Scale Waves 笛卡尔浅水模型：海啸和大规模波浪

Lecture: Shallow Water Modelling

Fluid dynamical models 流体动力学模型。我们将研究浅水模型。

1. Forces and motion 

– Newton’s laws, circular motion 

牛顿第一定律：假若施加于某物体的外力为零，则该物体的运动速度不变。（惯性定律）

牛顿第二定律：物体加速度的大小跟作用力成正比，跟物体的质量成反比，且与物体质量的倒数成正比；加速度的方向跟作用力的方向相同。第二定律指出了力的作用效果：力使物体获得加速度。

牛顿第三定律：相互作用的两个物体之间的作用力和反作用力总是大小相等，方向相反，作用在同一条直线上。第三定律揭示出力的本质：力是物体间的相互作用。

压力这个力F等于等于窗户上的压力变化ΔP乘以面积A

圆周运动：在圆周运动中，加速度等于-v²除以r，指向圆的中心。

Q:

Question 1: Which law does the following statement 'The net force acting on an object is equal to the product of mass and acceleration' represent?
A: Newton's second law
Question 2: Which of the following factors will usually give rise to forces?
A: pressure differences
Question 3: What does 'uniform circular motion' refer to?
A: motion in a circle at a constant tangential speed
Question 4: Which law can you use to calculate acceleration towards the centre of a circle made by an object on a string in a circular motion?
A: Pythagoras theorem

2. Equation of motion for fluid 

– Acceleration 

– Pressure gradient 

– Coriolis effect 

压力梯度力告诉我们地球将从高压区域移动到低压区域。但科里奥利力也告诉我们，空气的运动至少在北半球会向右弯曲。所以空气被推离高压区域，它将向右弯曲。如果空气向着低压区域移动，如图所示，它也将向右弯曲。它们都会向右弯曲。现在，如果我们连接这些箭头，我们就会得到围绕高压系统的气流是顺时针方向的。如果我们连接围绕这个低压系统的箭头，我们会得到围绕低压的气流是逆时针的。我们在大气中看到这一点，例如在气旋中，我们可以在卫星图上看到这种云带的模式。所以这是低压中心。这里发生的是云被拉入这个低压系统。但随着它这样做，空气向右弯曲。我们可以在一个螺旋中看到这个模式。

So, as I said before, the pressure gradient force tells us that the Earth will move from an area of high pressure to low pressure. But also, the Coriolis force tells us that the motion of the air will be bent to the right at least in the northern hemisphere. So the air's being pushed away from this high-pressure zone. It will bend to the right. And if air is moving towards this low-pressure zone, shown here, it will also bend to the right. They both bend to the right.

Now, if we join up the arrows, then we get that the motion of the air around a high-pressure system is in a clockwise direction. And if we join up the arrows around this low-pressure system, we get that the motion of the air around the low is anti-clockwise. And we see this in the atmosphere. We see this in cyclones, for instance, and we can see the pattern shown in this cloud banding here from a satellite. So this is a low-pressure centre. And what's happening here is that the clouds are being pulled into this low-pressure system. But the air is bending to the right as it does it. And we can see that pattern in a spiral.

我tm看不懂啊这是什么东西？？？？看看习题: 

Q:

1. In a scenario where the air is subject only the pressure gradient force, it will move from...

Answer: ... high pressure to low pressure in a straight line.

2. The following function f=2Ωsinɸ refers to...

Answer: ... the Coriolis parameter

3. In the northern hemisphere air flows clockwise around a high pressure system and anticlockwise...

Answer: ...around a low pressure system.

3. Balanced flow 

– Geostrophic wind 

由于科氏力需要平衡压力梯度力，它需要向相反的方向起作用，为了实现这一点地转风必须向沿等压面保持平行移动的风。地转风是描述在平衡流体运动中由于地球自转而产生的风向和风速的概念。

– Gradient wind 

在大气中平衡状态下，受到气压梯度力、科氏力和离心力（或向心力）综合作用而形成的一种风流。梯度风的产生是为了描述气体在不同纬度和高度上的平衡流动情况。它包含了气压梯度力趋向低压中心的作用、科氏力导致的风偏向以及离心力在水平方向上的影响。

梯度风的方向在低压系统中沿着等压线指向低压中心，在高压系统中则指向等压线远离高压中心的方向。梯度风的速度取决于气压梯度力、科氏参数和离心力，这三者的平衡使得大气在平流层内达到相对稳定的水平流动状态。

– Lows and highs 

上高下低风向下弯曲顺时针，反之逆时针

4. Vorticity 

正涡度是逆时针旋转，负涡度是顺时针旋转

5. Waves 

– Gravity waves and Tsunamis 

– Jets / barotropic instability 

– Rossby waves 

– Equatorial waves

Q: 

Question: In a scenario of air flowing around a low pressure system, in which direction does the pressure gradient force act? 

Answer: It acts towards the low (unlike the Coriolis force).

Question: Where are gravity waves formed? 

Answer: usually in the troposphere by frontal systems or by airflow over moountains.

Question: With regard to vorticity in a 3-D flow, is absolute vorticity conserved?  

Answer: No, it can change if the flow is stretched or compressed.

Question: What concept does this definition describe: the sum of relative vorticity, the curl of the flow and the spin of the planet?

Answer: It refers to absolute vorticity.

Computer Practice: Shallow Water Model

Default:

P1: Gravity waves: 重力波

重力波可以通过指定由非均匀高度场（除了左侧高斯流体斑点之外的平坦场）和静止风组成的初始条件激发The simplest wave modelled by the shallow water equations is the gravity wave. Gravity waves can be excited by specifying initial conditions consisting of a non-uniform height field (specifically in this example a field that is flat except for a Gaussian blob of fluid towards the left of the domain) and the winds at rest. The settings to use are as follows

0.5 day

The speed of the gravity waves is agree with the theoretical phase speed of c=根号gh

P2: Tsunami 海啸

模拟海啸之类的东西，其中的流体现在被视为具有不同底部地形的非常深的海洋。初始高度场中的高斯斑点可以被认为是地震引起的海洋表面的位移（尽管比真正的海啸大 3 级，以便我们不需要更改色标）。如果 H 太高以至于 h 变为负值，则模型将失败，因此我们添加高斯海山形式的地形，其最高点仅在地表以下 500 m：If we modify the previous experiment but add some orography then we model something like a tsunami where the fluid is now treated as a very deep ocean with a varying bottom topography. The Gaussian blob in the initial height field can be thought of as a displacement of the ocean surface due to an earthquake (although of much larger 3 magnitude than in a real tsunami in order that we don’t need to change the colour scale). The model will fail if H is so high that h goes negative, so we add topography in the form of a Gaussian sea mount which at its highest point is only 500 m below the surface:

When the gravity wave as it approaches shallower seas the tsunami slows down as it goes over the sea mount, because the effective h decreases the shape of the weave changes.

P3: Barotropic instability 正压不稳定性

正压不稳定性发生在经向（纬向）位涡流梯度改变符号的情况，可能发生在强切变的区域。比如喷流的边缘。

initial condition=zonal jet

initial condition= sharp shear

no random height noise

没有了随机噪声之后模型处于地转平衡也就是没有了加速度，随机噪声能够使模型略微失去平衡生成一些现象。Nothing happens when there is no random noise because the model is in geostrophic balance, which means there should be no acceleration! Random noise puts the model slightly out of balance, which is enough to do something interesting. 

P4: Jupiter's Great Red Spot

木星红斑是一个反气旋，为什么木星气旋能够存在如此之久。因为2D地转湍流的特性是小尺度结构往往更倾向于合并成更大的结构。在这个例子中，我们初始化高度场，使得从南到北移动时，经向（经度）风从东风变为西风，再变为东风，再变为西风，就像木星上的条件一样：In this example we initialize the height field such that as we move from south to north the zonal (longitudinal) wind switches from easterly to westerly to easterly to westerly, just like conditions on Jupiter:

P5: Orographic Rossby Wave 地形Rossby波（行星波 planetary waves）

地球大气层的一个显著特征是存在大尺度行星波，北半球这些波通常是由于空气流过喜马拉雅或者落基山脉激发的，模拟是空气流过孤立的高斯山的情况。In the northern hemisphere, these are excited by flow over major mountain ranges such as the Himalayas and the Rockies. Let's first simulate the flow over an isolated Gaussian mountain:

Gaussian mountain

在中纬度的均匀偏西风中，空气具有行星涡度（因为地球在旋转，即使空气相对于地面没有涡度）。如果空气被压缩，涡度就会减小。如果涡度起始为零，这必须意味着涡度变为负值——顺时针旋转，导致空气向南移动。如果空气被拉伸，涡度变为正值——逆时针旋转，导致空气向北移动。空气在越过山脉时被压缩，在山脉后被拉伸。In a uniform westerly in the mid-latitudes the air has planetary vorticity (because the planet is rotating, even if the air has no vorticity relative to the ground). If the air compresses the vorticity is reduced. If the vorticity starts at zero this must mean the vorticity goes negative – a clockwise rotation and the air travels south as a result. If the air is stretched the vorticity goes positive – an anti-clockwise rotation and the air travels north as a result. Air is compressed as it goes over the mountain and stretched as it comes off the mountain. 

f=0,beta=o

Simulation does not work because setting f to 0 means we divide by 0 when we set up the wind to be geostrophic.意味着建立风为地转平衡的时候我们除以0

f=1.e-4 beta=0

（考虑Rossby波从山脉后移动的恢复运动）。在这个模拟中，科里奥利参数不随纬度变化。这意味着行星涡度（地球的旋转）无法转化为空气的相对涡度。因此，这个Rossby波不如第一个案例强。这突显了Rossby波中的两个主要效应：（1）拉伸和压缩；（2）行星涡度转化为相对空气涡度。In this simulation the Coriolis parameter does not change with latitude. This means that planetary vorticity (the earth’s rotation) cannot be converted to relative vorticity of the air. So the Rossby wave is not as strong as in the first case. This highlights that there are two main effects in a Rossby wave (1) the stretching and compression; (2) the conversion of planetary vorticity to relative air vorticity. 

Earth Orography

air moving south downwind of the major mountain ranges – the Rockies and the Himalayas.

P6:  Equatorially trapped waves 赤道困波

赤道地区的波可能会被困在赤道附近：

P7: Equatorial Kelvin wave: 赤道开尔文波

在海洋中观察到的另一种赤道陷阱波是Kelvin波。这是一种发生在海洋混合层中的重力波；由于混合层较浅，它的传播速度足够慢，以使科里奥利力对其产生作用。具体而言，科里奥利力在北半球的运动方向右侧产生作用，在南半球的运动方向左侧产生作用，这意味着沿赤道传播的正高异常将受到科里奥利力的增强（因此寿命很长），但仅在它以一特定方向传播时（它向东传播！）。这就是赤道Kelvin波。

The Kelvin wave moves east!

P8: Weather forecast

模拟开始的时候出现的快速波动rapid fluctuations为重力波

1 Pa

= 1 N/m² = 1（kg·m·s^-2）/m² =1（kg·m^-1·s^-2）

= 0.01 mbar（毫巴）

= 10⁻⁵ bar（巴）

≈ 10⁻⁵ kgf/cm²（千克力/平方厘米）