Mathematical Modeling of the Water Cycle | 水循环的数学建模

📚 Mathematical Modeling of the Water Cycle | 水循环的数学建模

The water cycle is a continuous global process that moves water between the atmosphere, land, and oceans. Although it is typically studied in geography or environmental science, every stage of the cycle can be described and analysed using mathematics. In this A-Level revision article, we focus on the key mathematical tools — differential equations, balance equations, exponential models, and integration — that allow hydrologists to quantify the movement of water.

水循环是一个全球性的连续过程，它使水在大气、陆地和海洋之间不断运动。虽然水循环通常在地理或环境科学中学习，但它的每一个阶段都可以用数学来描述和分析。在这篇 A-Level 复习文章中，我们重点介绍关键数学工具——微分方程、平衡方程、指数模型和积分——它们使水文学家能够量化水的运动。

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1. The Water Balance Equation | 水量平衡方程

The foundation of hydrological modelling is the water balance equation. For any closed system, the change in water storage ΔS over a time period equals precipitation P minus evapotranspiration E minus runoff R. The algebraic statement is:

水文建模的基础是水量平衡方程。对于任何封闭系统，一段时间内的蓄水量变化 ΔS 等于降水量 P 减去蒸散量 E 再减去径流量 R。其代数表达式为：

ΔS = P − E − R

on is also applied over annual or monthly timescales, forming the basis of catchment water budget studies.

这个简单的线性方程可以重新排列以求解任何未知项。例如，如果我们测得 P、E 和河流蓄水量的变化，我们就可以计算径流量 R = P − E − ΔS。该方程也适用于年或月时间尺度，构成流域水量平衡研究的基础。

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2. Precipitation and Rainfall Intensity | 降水与降雨强度

Precipitation is the input to the water cycle. Rainfall intensity I is usually expressed in mm per hour and is defined as the derivative of rainfall depth D with respect to time t:

降水是水循环的输入项。降雨强度 I 通常以毫米/小时表示，定义为降雨深度 D 对时间 t 的导数：

I = dD/dt

If total rainfall is observed over a storm, the cumulative depth D(t) can be modelled using a polynomial or an exponential function. Then the average intensity is simply the total depth divided by the storm duration. Differentiation is therefore essential for finding instantaneous intensity values from cumulative rainfall data.

如果观测到一场风暴的总降雨量，累积深度 D(t) 可以用多项式或指数函数模拟。那么平均强度就是总深度除以风暴持续时间。因此，微分对于从累积降雨数据中求瞬时强度值至关重要。

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3. Evaporation: Exponential Decay Models | 蒸发：指数衰减模型

Evaporation removes water from lakes, soils, and vegetation. Water stored in a small pan exposed to constant conditions often decreases at a rate proportional to the remaining depth h. This gives a first-order differential equation:

蒸发会带走湖泊、土壤和植被中的水分。暴露在恒定条件小水盆中的蓄水深度 h 通常以减少速率与剩余深度成正比的方式下降。这给出一个一阶微分方程：

dh/dt = −k h

where k is a positive constant. Solving this equation gives h(t) = h₀ e^−kt onstant outflow. The half-life of the water in the pan can be found using t₁/₂ = ln 2 / k.

其中 k 为正常数。解此方程得到 h(t) = h₀ e^−kt，其中 h₀ 是初始深度。该指数衰减模型用于估算蒸发损失，也描述了恒定出流下水库存水的排空过程。盆中水的半衰期可由 t₁/₂ = ln 2 / k 求得。

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4. Infiltration and Darcy's Law | 入渗与达西定律

Infiltration is the process by which water enters the soil. A simplified empirical model is the Horton equation, which states that infiltration capacity f decreases exponentially with time:

入渗是水进入土壤的过程。一个简化的经验模型是霍顿方程，它表述入渗能力 f 随时间呈指数递减：

f(t) = f_c + (f₀ − f_c) e^{−α t}

Here f₀ is the initial infiltration rate, f_c is the constant final rate, and α is a decay constant. By integrating f(t) over a storm duration, we obtain the total infiltrated water volume. This use of integration demonstrates how continuous rates accumulate into total quantities.

这里 f₀ 是初始入渗率，f_c 是最终恒定入渗率，α 是衰减常数。通过对 f(t) 在风暴持续时间内积分，我们得到总入渗水量。积分的使用展示了连续速率如何累积为总量。

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5. Runoff Calculation with Integration | 用积分计算径流

Runoff occurs when precipitation exceeds infiltration and storage capacity. If the rainfall intensity I(t) and infiltration capacity f(t) are known, the runoff rate Q(t) at any instant can be approximated as:

当降水超过入渗和蓄水能力时，就会产生径流。如果已知降雨强度 I(t) 和入渗能力 f(t)，任一瞬间的产流率 Q(t) 可近似为：

Q(t) = max(0, I(t) − f(t))

The total runoff volume over the storm is the integral of Q(t) with respect to time. In practical modelling, numerical integration methods such as the trapezium rule or Simpson's rule are used when the functions are too complex for analytical integration.

整个风暴期间的总径流量是 Q(t) 对时间的积分。在实际建模中，当函数过于复杂而无法解析积分时，会使用数值积分方法，如梯形法则或辛普森法则。

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6. Groundwater Flow and Continuity | 地下水流动与连续性

Groundwater moves slowly through porous rock. Darcy's law states that the specific discharge q (volume per unit area per time) is proportional to the hydraulic gradient:

地下水在孔隙岩石中缓慢流动。达西定律指出，比流量 q（单位面积单位时间的体积）与水力梯度成正比：

q = −K (dh/dl)

ivity, h is the hydraulic head, and l is the distance along the flow path. Combining Darcy' e flow in a confined aquifer can be modelled by d²h/dx² = 0, giving a linear head profile.

这里 K 是水力传导系数，h 是水头，l 是沿流动路径的距离。将达西定律与连续性方程结合，得到描述地下水水位的微分方程。在一维情况下，承压含水层中的稳态流可以用 d²h/dx² = 0 建模，得到线性水头分布。

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7. Residence Time and Flow Rates | 滞留时间与流速

Residence time measures how long a water molecule stays in a reservoir such as a lake, river, or aquifer. It is calculated by dividing the reservoir volume V by the average flow rate Q:

滞留时间衡量一个水分子在湖泊、河流或含水层等蓄水体内停留的时间。它通过蓄水体积 V 除以平均流速 Q 来计算：

T = V / Q

Although this appears to be a simple arithmetic formula, it has deep connections to probability: the distribution of residence times in a well-mixed reservoir follows an exponential distribution. The mean residence time equals the reciprocal of the fractional loss rate k, connecting algebraic ratios with exponential decay models.

尽管这看起来是一个简单的算术公式，但它与概率有深刻联系：完全混合蓄水体内滞留时间的分布服从指数分布。平均滞留时间等于分数损失率 k 的倒数，从而将代数比值与指数衰减模型联系起来。

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8. Differential Equations for Reservoir Storage | 蓄水量的微分方程

The dynamic change in water storage in a reservoir can be expressed as a differential equation. If inflow is I(t) and outflow is O(t), the storage S changes according to:

蓄水体内蓄水量的动态变化可以表示为微分方程。如果入流为 I(t)，出流为 O(t)，蓄水量 S 按如下方程变化：

dS/dt = I(t) − O(t)

s technique. For a constant inflow I₀, the general solution is S(t) = I₀/k + (S₀ − I₀/k) e^−kt, showing convergence to equilibrium.

在许多情况下，出流与蓄水量成正比，即 O(t) = k S。方程变为 dS/dt = I(t) − k S，这是一个一阶线性微分方程。求解此方程需要积分因子，这是 A-Level 数学的关键技巧。对于恒定入流 I₀，通解为 S(t) = I₀/k + (S₀ − I₀/k) e^−kt，显示出向平衡收敛的趋势。

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9. Numerical Methods and Hydrological Models | 数值方法与水文模型

ological systems are too complex for exact analytical solutions. Engineers often use Euler' mate solutions of differential equations. For the reservoir equation dS/dt = f(t,S), Euler's step formula is:

真实水文系统过于复杂，无法精确解析求解。工程师通常使用欧拉方法或龙格-库塔方法来近似求解微分方程。对于蓄水方程 dS/dt = f(t,S)，欧拉的步进公式为：

S_n+1 = S_n + h f(t_n, S_n)

where h is the time step. Numerical accuracy improves as h decreases, but computational cost increases. This trade-off is a central idea in applied mathematics and is tested in A-Level numerical methods questions.

其中 h 是时间步长。随着 h 减小，数值精度提高，但计算成本增加。这种权衡是应用数学的核心思想，也是 A-Level 数值方法问题中的考点。

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10. The Unit Hydrograph and Convolution | 单位过程线与卷积

The unit hydrograph is a classic hydrological tool that predicts runoff from rainfall. It is the response of a catchment to one unit of effective rainfall. For a time-varying rainfall excess x(t), the resulting runoff y(t) is approximated by the convolution integral:

单位过程线是预测降雨径流的经典水文工具。它是流域对一单位净雨量的响应。对于随时间变化的净雨 x(t)，产生的径流 y(t) 可近似用卷积积分表示：

y(t) = ∫₀ᵗ u(τ) x(t − τ) dτ

Here u(τ) is the unit hydrograph. In discrete form, this becomes a sum of products, which is related to matrix multiplication. Understanding convolution strengthens the link between integration, linear systems, and real-world water forecasting.

这里 u(τ) 是单位过程线。在离散形式下，这变成乘积之和，与矩阵乘法有关。理解卷积强化了积分、线性系统与真实水文预报之间的联系。

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11. Climate Change and Mathematical Projections | 气候变化与数学预测

Mathematical models are essential for predicting how the water cycle responds to climate change. Temperature increases affect evaporation rates through the Clausius-Clapeyron relation, approximately increasing atmospheric moisture capacity by 7% per degree Celsius. This can be written as:

数学模型对于预测水循环如何响应气候变化至关重要。温度升高通过克劳修斯-克拉佩龙关系影响蒸发速率，大气水汽容量大约每摄氏度增加 7%。这可以写成：

e_s(T) = e_s(T₀) × 1.07^{T − T₀}

where e_s is the saturation vapour pressure and T is temperature in degrees Celsius. This exponential model allows hydrologists to estimate changes in extreme rainfall intensity and flood risk — a direct application of the exponential functions studied in A-Level mathematics.

其中 e_s 是饱和水汽压，T 是摄氏温度。这个指数模型使水文学家能够估算极端降雨强度和洪水风险的变化——这是 A-Level 数学中指数函数的直接应用。

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12. Conclusion: Mathematics as the Language of the Water Cycle | 结论：数学是水循环的语言

Every stage of the water cycle — from evaporation and condensation to infiltration, runoff, and groundwater flow — can be expressed mathematically. Differential equations describe dynamic changes, integration accumulates rates into volumes, and exponential functions model decay and growth. These tools are not only examined in A-Level mathematics but also underpin cutting-edge hydrological research and water resource management.

水循环的每一个阶段——从蒸发、凝结到入渗、径流和地下水流动——都可以用数学表达。微分方程描述动态变化，积分将速率累加为总量，指数函数模拟衰减和增长。这些工具不仅在 A-Level 数学中考查，而且支撑着前沿水文研究和水资源管理。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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