Urban Climate: Mathematical Modelling and Data Analysis | 城市气候：数学建模与数据分析

📚 Urban Climate: Mathematical Modelling and Data Analysis | 城市气候：数学建模与数据分析

Urban climate is the local modification of regional climate caused by buildings, roads, industry and other human activities. In A-level mathematics, we can model these effects using energy balance equations, statistical correlation and regression, and simple differential equations. This article links key ideas from Edexcel Applied Mathematics and Statistics to the study of urban heat islands, air pollution and wind patterns.

城市气候是指建筑物、道路、工业和其它人类活动对区域气候造成的局部改变。在 A-level 数学中，我们可以用能量平衡方程、统计相关与回归以及简单微分方程来模拟这些影响。本文将 Edexcel 应用数学与统计学的核心概念同城市热岛、空气污染和风场研究联系起来。

1. Urban Climate and Mathematical Variables | 城市气候与数学变量

To model urban climate, we define variables such as urban air temperature Tᵤ, rural air temperature Tᵣ, wind speed u, surface albedo α, and net radiation Q*. These variables allow us to write equations and compare different locations or times using quantitative methods.

为了模拟城市气候，我们定义一些变量，如城市气温 Tᵤ、乡村气温 Tᵣ、风速 u、地表反照率 α 和净辐射 Q*。这些变量使我们能够写出方程，并用定量方法比较不同地点或时段。

A key measure is the urban heat island intensity, written as the temperature difference between an urban site and a nearby rural site. It is positive especially at night when concrete releases stored heat more slowly than vegetation does.

一个关键指标是城市热岛强度，它表示城市站点与邻近乡村站点之间的气温差。夜间这一数值通常为正，因为混凝土比植被更缓慢地释放储存的热量。

ΔT = Tᵤ − Tᵣ

2. The Urban Heat Island Equation | 城市热岛方程

The simplest mathematical model treats the night-time urban heat island as a linear function of the logarithm of city population. This empirical relationship was first studied by Oke: larger cities tend to have larger ΔT.

最简单的数学模型将夜间城市热岛表示为城市人口对数的线性函数。这一经验关系最早由 Oke 研究：城市规模越大，ΔT 往往越大。

ΔT = a + b log₁₀(P)

Here P is population, and a and b are constants found by linear regression. For example, some data sets give b around 2–4 °C per unit increase in log₁₀(P). You can use a scientific calculator or spreadsheet to estimate a and b from paired data.

其中 P 是人口，a 和 b 是通过线性回归求出的常数。例如，有些数据集给出的 b 约为每增加一个 log₁₀(P) 单位，热岛强度升高 2–4 °C。你可以使用科学计算器或电子表格根据配对数据来估计 a 和 b。

This model is useful because it turns a physical observation into a predictive equation. It also links to the statistics topic of least squares regression.

这个模型很有用，因为它把物理观测转化为预测方程。它还与统计学中的最小二乘回归主题直接相关。

3. Surface Energy Balance | 地表能量平衡

Urban surfaces gain and lose energy through radiation, convection, evaporation and conduction. The energy balance can be written as a simple conservation equation.

城市表面通过辐射、对流、蒸发和传导获得和失去能量。能量平衡可以写成一个简单的守恒方程。

Q* = Q_H + Q_E + Q_G

Here Q* is net radiation, Q_H is sensible heat transferred to the air, Q_E is latent heat used in evaporation, and Q_G is heat stored in the ground and buildings. In cities, Q_E is lower because there is less vegetation, so a larger fraction of Q* becomes Q_H, warming the air.

其中 Q* 是净辐射，Q_H 是传给空气的感热，Q_E 是蒸发所用的潜热，Q_G 是储存在地面和建筑物中的热量。在城市中，由于植被较少，Q_E 较低，因此 Q* 中有更大比例转化为 Q_H，使空气升温。

Mathematically, if Q_E decreases while Q* and Q_G stay roughly constant, Q_H must increase to maintain the balance. This explains why urban air is warmer.

从数学上看，如果 Q* 和 Q_G 大致不变，而 Q_E 减少，那么 Q_H 必须增加才能维持平衡。这就解释了为什么城市空气更热。

4. Albedo and Net Radiation | 反照率与净辐射

Albedo α is the fraction of incoming shortwave radiation reflected by a surface. It is a ratio, so it has no units and lies between 0 and 1.

反照率 α 是地表反射的入射短波辐射所占的比例。它是一个比值，因而没有单位，取值在 0 到 1 之间。

α = K↑ / K↓

Urban materials such as asphalt have low albedo, typically 0.04–0.10, while fresh snow can have an albedo above 0.80. Low albedo means more absorbed solar energy, which can raise daytime surface temperatures.

沥青等城市材料的反照率较低，通常为 0.04–0.10，而新雪的反照率可超过 0.80。反照率低意味着吸收的太阳能更多，从而使白天表面温度升高。

Surface \| 表面	Typical Albedo α
Fresh asphalt \| 新沥青	0.04–0.08
Concrete \| 混凝土	0.10–0.25
Grass \| 草地	0.15–0.30
Fresh snow \| 新雪	0.80–0.90

In mathematical terms, increasing albedo leads to a lower absorbed shortwave flux K*, where K* = (1 − α) K↓. Urban planners use this idea when they design cool roofs with higher albedo.

从数学上讲，提高反照率会降低吸收的短波通量 K*，其中 K* = (1 − α) K↓。城市规划者在设计具有更高反照率的冷屋顶时，正是利用了这一点。

5. Regression Models for Heat Island Intensity | 热岛强度的回归模型

You may be asked to interpret a scatter graph of heat island intensity against city population, wind speed or green space percentage. Use the product moment correlation coefficient to measure linear association.

你可能会遇到热岛强度与城市人口、风速或绿地百分比之间的散点图分析题。使用积矩相关系数来衡量线性相关程度。

r = Σ(xᵢ−x̄)(yᵢ−ȳ) / √[ Σ(xᵢ−x̄)² Σ(yᵢ−ȳ)² ]

A negative r between wind speed and ΔT means that stronger winds reduce heat island intensity by mixing warmer urban air with cooler rural air. A positive r between built-up area and ΔT means that more buildings are associated with a stronger heat island.

风速与 ΔT 之间的 r 为负，说明更强的风通过混合较暖的城市空气与较冷的乡村空气来减弱热岛强度。建筑面积与 ΔT 之间的 r 为正，说明建筑越多，热岛强度往往越大。

When using regression lines, remember the difference between interpolation and extrapolation. Predicting ΔT for a city whose population lies outside the data range is less reliable.

在使用回归线时，要注意内插与外推的区别。如果城市人口超出数据范围，那么用它来预测 ΔT 的可靠性较低。

6. Air Pollution: Concentration and Dispersion | 空气污染：浓度与扩散

Urban areas have higher emissions from traffic and industry. A simple model for pollutant concentration downwind of a source is based on the idea that concentration decreases as distance increases.

城市地区来自交通和工业的排放物较多。一个简单的源下风向污染物浓度模型基于这样的思想：浓度随距离增加而下降。

C(x) = C₀ e^(−kx)

Here C₀ is the initial concentration, x is distance from the source, and k is a positive decay constant. This exponential model is similar to radioactive decay and other A-level applications.

其中 C₀ 是初始浓度，x 是距污染源的距离，k 是一个正的衰减常数。这个指数模型与放射性衰变以及其它 A-level 应用类似。

Wind speed u also affects concentration. If emission rate Q is constant, average concentration C near a line source such as a road can be approximated by C = Q / (u × W × H), where W is width and H is mixing height. Doubling wind speed roughly halves the concentration.

风速 u 也会影响浓度。如果排放速率 Q 恒定，道路等线源附近的平均浓度 C 可近似表示为 C = Q / (u × W × H)，其中 W 为宽度，H 为混合高度。风速加倍大约会使浓度减半。

7. Rainfall Trends and Statistical Testing | 降雨趋势与统计检验

Some studies suggest that cities may enhance rainfall downwind because heat and aerosol particles affect cloud formation. To test such a claim, we can compare monthly rainfall at urban and rural stations using a paired t-test.

一些研究表明，城市可能使下风向降雨增加，因为热量和气溶胶粒子会影响云的形成。为了检验这种说法，我们可以使用配对 t 检验比较城市和乡村站点的月降雨量。

t = (x̄_d) / (s_d / √n)

Here x̄_d is the mean difference between paired monthly rainfall values, s_d is the standard deviation of differences, and n is the number of paired observations. If the calculated t exceeds the critical value at the 5% significance level, we reject the null hypothesis.

其中 x̄_d 是成对月降雨量差值的平均值，s_d 是差值的标准差，n 是成对观测值的数量。如果计算出的 t 值超过 5% 显著性水平下的临界值，我们就拒绝原假设。

You should always state the null hypothesis, such as H₀: μ_d = 0, and the alternative hypothesis H₁: μ_d > 0 if you are testing for an urban increase.

你应该始终说明原假设，例如 H₀: μ_d = 0，以及在检验城市降雨增加时的备择假设 H₁: μ_d > 0。

8. Wind Speed and Urban Roughness | 风速与城市粗糙度

Tall buildings increase surface roughness, slowing the wind near street level. Mean wind speed at height z can be modelled with a power law.

高层建筑增加了地表粗糙度，使近地面风速减慢。高度 z 处的平均风速可以用幂律来模拟。

u(z) = u_ref × (z / z_ref)^α

Here u_ref is the wind speed at a reference height z_ref, and α is the roughness exponent. Urban areas have larger α, so wind speed increases more slowly with height than in open country.

其中 u_ref 是参考高度 z_ref 处的风速，α 是粗糙度指数。城市地区的 α 较大，因此风速随高度增加的速度比开阔乡村更慢。

From a statistical viewpoint, wind data are often skewed. You may need to calculate the median rather than the mean, or use a logarithmic transformation before applying normal-distribution methods.

从统计角度看，风速数据往往呈偏态分布。你可能需要计算中位数而不是均值，或者在应用正态分布方法之前使用对数变换。

9. Urban Climate Data Collection | 城市气候数据收集

Fieldwork may involve measuring air temperature at fixed points along an urban-rural transect. You can then calculate descriptive statistics such as mean, median, range, interquartile range and standard deviation.

实地考察可能包括沿城乡样带在固定点测量气温。然后你可以计算描述性统计量，如均值、中位数、极差、四分位距和标准差。

When comparing two samples, use box plots to show differences in central tendency and spread. A large interquartile range for urban sites may indicate microclimatic variation caused by shade, traffic and building materials.

比较两个样本时，可以使用箱线图显示集中趋势和离散程度的差异。城市站点四分位距较大，可能表明阴影、交通和建筑材料引起了微气候差异。

Remember that temperature measurements have uncertainty. If the thermometer reads to ±0.5 °C, then a reported heat island intensity of 2.0 °C has a maximum possible error of 1.0 °C when two readings are subtracted.

请记住，温度测量存在不确定性。如果温度计的读数精确到 ±0.5 °C，那么当两个读数相减时，报告的热岛强度 2.0 °C 的最大可能误差为 1.0 °C。

10. Management Strategies: Cost-Benefit Analysis | 治理策略：成本效益分析

Urban climate management includes green roofs, cool pavements, tree planting and better ventilation. Mathematical models can compare the estimated cooling benefit with the initial and maintenance costs.

城市气候治理包括绿色屋顶、冷路面、植树和改善通风。数学模型可以比较估算的降温收益与初始及维护成本。

Net Benefit = Σ (Bₜ − Cₜ) / (1 + r)^t

This is a discounted cash flow formula. Bₜ and Cₜ are benefits and costs in year t, and r is the discount rate. A positive net benefit supports the investment.

这是一个折现现金流公式。Bₜ 和 Cₜ 是第 t 年的收益和成本，r 是折现率。净收益为正说明该投资是合理的。

You can also calculate the percentage reduction in heat island intensity after a mitigation scheme: ((ΔT_before − ΔT_after) / ΔT_before) × 100%.

你还可以计算缓解方案实施后热岛强度下降的百分比：((ΔT_before − ΔT_after) / ΔT_before) × 100%。

11. Case Study: London’s Urban Heat Island | 案例研究：伦敦热岛

London’s urban heat island is strongest on clear, calm nights. The centre can be 6–8 °C warmer than surrounding rural areas, especially after a hot day when buildings release stored heat.

伦敦的城市热岛在晴朗无风的夜晚最强。市中心可比周边乡村地区温度高 6–8 °C，尤其是在炎热的一天之后，建筑物释放储存的热量时更为明显。

Using the equation ΔT = a + b log₁₀(P), London’s population of about 9 million gives log₁₀(P) ≈ 6.95. Typical model estimates then give ΔT values close to observed values of 6–8 °C.

使用方程 ΔT = a + b log₁₀(P)，伦敦约 900 万人口对应的 log₁₀(P) ≈ 6.95。典型模型估计给出的 ΔT 值接近观测值 6–8 °C。

London has also introduced the Ultra Low Emission Zone, which reduces traffic emissions. Statistical analysis of nitrogen dioxide data before and after the policy uses two-sample t-tests or confidence intervals to test for significant improvement.

伦敦还设立了超低排放区以减少交通排放。对政策实施前后二氧化氮数据进行统计分析时，可使用双样本 t 检验或置信区间来检验是否有显著改善。

12. Exam Tips for Mathematical Analysis | 数学分析的考试技巧

When answering an urban climate question in an applied mathematics paper, always define your variables, show your formula, substitute values accurately and give units. Round final answers to an appropriate degree of accuracy.

在应用数学试卷中回答城市气候问题时，务必定定义变量、写出公式、准确代入数值并标明单位。最终答案应四舍五入到适当的精确度。

Define ΔT as urban minus rural temperature, not the reverse.
Check whether a correlation means causation; urban heat and population may both be linked to land use.
Use a calculator for regression and t-tests, but write down the hypotheses and critical values.
In exponential decay models, label C₀ and k clearly and state their units.

将 ΔT 定义为城市温度减乡村温度，不要弄反。
注意相关不代表因果；城市热量和人口可能都与土地利用有关。
使用计算器进行回归和 t 检验，但要写出假设和临界值。
在指数衰减模型中，清楚标注 C₀ 和 k 并说明单位。

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Urban Climate: Mathematical Modelling and Data Analysis | 城市气候：数学建模与数据分析