📚 Year 13 WJEC Geography: Formula & Theorem Quick Reference Guide | Year 13 WJEC 地理：公式定理速查手册
This quick reference guide brings together the essential quantitative relationships, physical laws and conceptual models required for the WJEC A Level Geography specification. Whether you are analysing drainage basin dynamics, coastal energy or urban hierarchies, mastering these formulas and theorems will strengthen your data-response answers and help you explain geographical patterns with precision. Each entry is presented with a concise explanation, key variables and typical applications in the context of Year 13 topics.
这本速查手册汇集了 WJEC A Level 地理考试所必需的核心定量关系、物理定律和概念模型。无论你在分析流域动态、海岸能量还是城市等级体系，掌握这些公式和定理都能让你的数据分析题答案更有力，帮助你精准地解释地理格局。每一条目都附有简明解释、关键变量及其在 Year 13 主题中的典型应用。
1. Drainage Basin Hydrology: Discharge and Water Balance | 流域水文学：流量与水平衡
Stream discharge (Q) is the volume of water passing a cross‑section per unit time. It is the fundamental measure of a river’s flow magnitude, expressed as Q = A × V, where A is the cross‑sectional area (m²) and V is the mean velocity (m s⁻¹). In fieldwork and exam calculations, you often derive V from float or flow‑meter measurements and multiply by the wetted area.
河流流量 (Q) 是单位时间内通过某一断面的水体体积，是衡量河流水量的基本指标，公式为 Q = A × V，其中 A 为过水断面面积 (m²)，V 为平均流速 (m s⁻¹)。在野外作业和考试计算中，通常通过浮标或流速仪测出 V，再乘以湿周面积求得流量。
The water balance equation for a drainage basin over a given time period is: Precipitation (P) = Runoff (Q) + Evapotranspiration (E) ± Change in storage (ΔS). This identity underpins the study of catchment hydrology and flood risk, allowing you to evaluate how human interventions – such as urbanisation or afforestation – alter the partitioning of rainfall.
流域在一定时段内的水平衡方程为：降水量 (P) = 径流量 (Q) + 蒸散量 (E) ± 储水变化量 (ΔS)。这一恒等式是流域水文学与洪水风险研究的基础，可用于评价城市化、植树造林等人类活动如何改变降水的分配。
Q = A × V P = Q + E ± ΔS
2. Manning’s Equation for Channel Flow | 曼宁公式
Manning’s equation estimates mean velocity in open channels where flow is uniform. It is widely used to assess channel efficiency and flood conveyance: V = (1/n) × R2/3 × S1/2, or in Unicode form V = (1/n) × R²⁄³ × S¹⁄². The variable n is Manning’s roughness coefficient (dimensionless but depends on unit system), R is the hydraulic radius (m), and S is the water‑surface slope (dimensionless, often approximated by channel gradient).
曼宁公式用于估算明渠均匀流中的平均流速，被广泛用于评估河道输水效率和洪水过流能力：V = (1/n) × R²⁄³ × S¹⁄²。其中 n 为曼宁粗糙系数，R 为水力半径 (m)，S 为水面比降（常近似为河道坡降）。
Roughness n encapsulates the effect of bed material, vegetation and channel irregularities. In exam contexts, you may be asked to predict how a change in channel management – e.g. removing bank vegetation (lower n) – would increase velocity and potentially heighten erosion. Manning’s equation also illustrates why deeper, hydraulically efficient channels transport flow faster under the same slope.
粗糙系数 n 综合反映了河床物质、植被和河道不规则性的影响。考试中可能要求你预测河道管理的改变——例如清除岸坡植被（n 降低）——会如何增大流速并可能加剧侵蚀。曼宁公式还说明，为何在相同比降下，水力效率更高的深槽型河道输水更快。
3. Hydraulic Radius and Efficiency | 水力半径与效率
The hydraulic radius (R) is defined as the ratio of cross‑sectional area to wetted perimeter: R = A / P. It is a key control on flow resistance; channels with a higher R (for a given cross‑sectional area) have less friction from the bed and banks, and thus a greater conveyance capacity. A wide, shallow channel has a low hydraulic radius, while a deep, narrow channel has a high one – until bank friction becomes dominant.
水力半径 (R) 定义为过水断面面积与湿周的长度之比：R = A / P。它是控制水流阻力的关键参数；在面积相同的情况下，R 越大，河床与岸壁的摩擦越小，输水能力越强。宽浅河道的水力半径偏小，而深窄河道则偏大——直到岸壁摩擦占据主导。
When assessing river management or flood defence schemes, planners often aim to modify the channel shape to increase R, thereby allowing the same discharge to pass at a lower flood stage. The concept also links to Manning’s equation: R raised to the power ²⁄₃ means that even small increases in depth relative to width can significantly raise velocity.
在评估河道治理或防洪方案时，规划者往往通过改变断面形态来提高水力半径，从而以较低水位宣泄同等流量。这一概念与曼宁公式紧密相连：R 的 ²⁄₃ 次方意味着，相对于宽度即使小幅增加水深也能显著提高流速。
4. Gradient and Sinuosity | 河道坡降与弯曲度
Channel gradient (S) is the vertical drop over a horizontal reach, often expressed as a dimensionless ratio or as m/km. Gradient influences potential energy available for sediment transport and is a central variable in erosion formulae. Sinuosity (SI) compares the actual length of a channel along its bends to the straight‑line valley distance: SI = channel length / valley length. A perfectly straight river has SI = 1; a meandering river has SI > 1.5.
河道坡降 (S) 是某一河段两端的高差与水平距离之比，常以无量纲比值或 m/km 表示。坡降决定了可用于泥沙搬运的势能，是侵蚀公式中的核心变量。弯曲度 (SI) 则对比河道沿曲折走向的实际长度与谷底线直线距离：SI = 河道长度 / 河谷长度。完全顺直的河流 SI=1，曲流河 SI>1.5。
These two variables interact to produce characteristic channel patterns. High‑gradient upland rivers often have low sinuosity due to coarse bedload and confined valleys; lower‑gradient lowland rivers can develop high sinuosity as lateral erosion dominates. Changes in sinuosity through channelisation reduce travel time, affect flood peak timing and frequently have unintended geomorphic consequences downstream.
坡降与弯曲度相互作用，形成具特色的河道形态。高坡降的山地河因粗粒床沙和峡谷约束，弯曲度通常较低；低坡降的平原河则因侧向侵蚀活跃而形成高弯曲度的河曲。渠化工程降低弯曲度会缩短传播时间、影响洪峰时机，并常在下游引发意料之外的地貌响应。
5. Stream Power and Erosion | 河流功率与侵蚀
Total stream power (Ω) expresses the energy available for geomorphic work per unit channel length: Ω = ρ × g × Q × S, where ρ is the density of water (1000 kg m⁻³), g is gravitational acceleration (9.81 m s⁻²), Q is discharge (m³ s⁻¹) and S is the channel slope. Alternatively, unit stream power (ω) is the power per unit bed area: ω = Ω / bed width.
总河流功率 (Ω) 表示单位河长上用于地貌塑造的有效能量：Ω = ρ × g × Q × S。其中 ρ 为水体密度 (1000 kg m⁻³)，g 为重力加速度 (9.81 m s⁻²)，Q 为流量 (m³ s⁻¹)，S 为河道坡降。单位河流功率 (ω) 则是单位河床面积的功率：ω = Ω / 河宽。
Stream power is frequently used to define thresholds for sediment transport and channel adjustment. When Ω exceeds a critical value, erosion or bed‑scour may occur; when it drops below a deposition threshold, sediments settle. Urbanisation that increases peak Q and steepens S locally can push a formerly stable reach into an erosional phase, requiring Grade Control Structures to dissipate energy.
河流功率常用于界定泥沙搬运和河道调整的阈值。当 Ω 超过临界值，可能发生侵蚀或河床冲刷；当它低于堆积阈值时，泥沙开始沉降。城市化增加洪峰流量并使局部坡降变陡，会把原本稳定的河段推入侵蚀阶段，此时往往需要设置消能设施（Grade Control Structures）以消耗能量。
6. Hack’s Law | 哈克定律
Hack’s law describes the empirical relationship between main‑channel length (L) and drainage‑basin area (A): L = c × Ah. The exponent h typically lies between 0.5 and 0.6 in unglaciated terrain, while the coefficient c varies with lithology and climate. In an ideal rectangular basin, h would be 0.5, but elongation and topography push it higher.
哈克定律描述了干流河道长度 (L) 与流域面积 (A) 之间的经验关系：L = c × Ah。在非冰川地区，指数 h 通常介于 0.5–0.6 之间，系数 c 则随岩性和气候而变化。理想矩形流域的 h 值为 0.5，但流域形态拉长和地形差异会使指数增高。
For Year 13, Hack’s law helps explain drainage‑basin morphometry and the scaling of fluvial processes. You might use it to compare basins of different sizes and infer how flood‑routing times change. It also feeds into landscape evolution models, linking the geometry of drainage networks to the underlying geomorphic factors that shape them.
在 Year 13 课程中，哈克定律有助于解释流域形态计量特征以及河流过程的尺度效应。你可以用它来比较不同大小的流域，推断洪水演进时间的变化。该定律还融入景观演化模型，将水系几何形态与控制其发展的根本地貌因子联系在一起。
7. Coastal Wave Energy and Longshore Drift | 海岸波浪能与沿岸漂移
Wave power (P) per metre of wave crest in deep water is given by: P ≈ 0.5 × (ρ × g² × H² × T) / 16π, which simplifies to approximately 0.5 × H² × T kW m⁻¹ when using metric units and standard values. Here H is significant wave height (m) and T is wave period (s). This energy drives processes of erosion, transport and beach building.
深水条件下每米波峰线上的波浪功率 (P) 可表示为：P ≈ 0.5 × (ρ × g² × H² × T) / 16π，使用公制单位和标准数值时可简化为约 0.5 × H² × T kW m⁻¹。式中 H 为有效波高 (m)，T 为波浪周期 (s)。该能量驱动海岸侵蚀、输沙及海滩塑造过程。
Longshore drift volume (Qls) depends on the alongshore component of wave energy: Qls = K × P × sin θb, where θb is the angle of wave break relative to the shoreline, and K is a site‑specific coefficient. Constructing groynes or altering offshore gradients changes this balance and leads to erosion/accretion patterns visible in spit and tombolo formation.
沿岸漂移量 (Qls) 取决于波浪能量的沿岸分量：Qls = K × P × sin θb，其中 θb 为波浪破碎方向与海岸线的夹角，K 为地点特定系数。建造丁坝或改变近岸坡度会打破这一平衡，从而导致在沙嘴和连岛坝的形态塑造中可以观察到的侵蚀或淤积格局。
8. Glacial Mass Balance and Flow | 冰川物质平衡与流动
Glacier mass balance (B) over a year is the algebraic sum of accumulation (snowfall, avalanches) and ablation (melting, sublimation, calving): B = accumulation − ablation. Positive mass balance in the accumulation zone drives ice‑flow towards the ablation zone, where net loss dominates. The equilibrium‑line altitude (ELA) separates the two zones and its rise over time flags glacial retreat.
冰川的年物质平衡 (B) 是积累（降雪、雪崩）与消融（融化、升华、崩解）的代数和：B = 积累量 − 消融量。积累区的正物质平衡驱动冰体向消融区流动，后者以净损失为主。平衡线高度 (ELA) 划开这两部分，其逐年升高标志着冰川退缩。
Ice velocity is often approximated through Glen’s flow law: ε̇ = A × τn, where ε̇ is the strain rate, A is a temperature‑dependent constant, τ is the applied shear stress and n is typically around 3. This non‑linear relationship explains why small increases in basal shear stress can accelerate ice‑discharge tenfold, a critical factor in dynamic responses to climate warming.
冰川流速常通过格伦流变定律近似：ε̇ = A × τn，式中 ε̇ 为应变速率，A 为温度相关常数，τ 为施加的剪切应力，n 通常约为 3。这一非线性关系解释了为何基底剪切应力的小幅增加即可使冰流量加速十倍，这是冰川对气候变暖发生动态响应的关键因子。
9. Carbon Cycle: Residence Time | 碳循环：停留时间
Residence time (T) of carbon in a given reservoir is the average time a carbon atom spends there, estimated as T = mass of carbon in reservoir (Pg) / total flux out (Pg yr⁻¹). This concept applies to the atmosphere, biosphere, hydrosphere and lithosphere. For instance, the residence time of CO₂ in the atmosphere is about 4–5 years for individual molecules but much longer for the bulk CO₂ anomaly due to rapid recycling with the ocean.
碳在特定库中的停留时间 (T) 是指一个碳原子在库内的平均驻留时间，可用 T = 库中碳质量 (Pg) / 总输出通量 (Pg yr⁻¹) 估算。这一概念适用于大气圈、生物圈、水圈和岩石圈。例如，单个 CO₂ 分子在大气中的停留时间约为 4–5 年，但由于与海洋的快速交换，大气中 CO₂ 异常的总停留时间要长得多。
Residence time links directly to the sensitivity of different stores to anthropogenic emissions. The lithosphere – with a residence time of millions of years – acts as a slow‑turnover sink, whereas the biosphere (decades) and atmosphere (years to centuries) respond quickly to changes in flux. In the WJEC Carbon Cycle topic, you are expected to use residence‑time calculations to evaluate the effectiveness of carbon sequestration strategies.
停留时间直接关系到不同碳库对人为排放的敏感程度。停留时间长达数百万年的岩石圈是一个周转缓慢的碳汇，而生物圈（几十年）和大气圈（数年至数百年）则对通量变化反应迅速。在 WJEC 碳循环主题中，你需要运用停留时间计算来评价碳封存策略的有效性。
10. Demographic Rates and Population Change | 人口统计率与人口变化
The basic demographic equation states: Population change = (Births − Deaths) + (Immigration − Emigration). From this, vital rates are derived: Crude Birth Rate (CBR) = (Total births / Mid‑year population) × 1000; Crude Death Rate (CDR) = (Total deaths / Mid‑year population) × 1000; fertility and mortality indices are all expressed per 1000 persons for comparability.
基本人口统计等式为：人口变化 = (出生数 − 死亡数) + (迁入数 − 迁出数)。由此衍生出各项比率：粗出生率 (CBR) = (出生总数 / 年中人口) × 1000；粗死亡率 (CDR) = (死亡总数 / 年中人口) × 1000；生育率和死亡率指数为了可比性均以每千人表示。
Replacement‑level fertility is approximately 2.1 children per woman in developed countries, accounting for infant and child mortality. The Total Fertility Rate (TFR) is a synthetic cohort measure: TFR = sum of age‑specific fertility rates (ASFR) × age‑interval width. These indicators allow you to analyse population momentum and forecast growth trajectories under different development scenarios.
在发达国家，更替水平的生育率约为每名妇女生育 2.1 个孩子，这与婴幼儿死亡率有关。总和生育率 (TFR) 是一项合成队列指标：TFR = Σ (年龄别生育率 × 年龄段宽度)。借助这些指标可以分析人口惯性，预测不同发展情景下的人口增长轨迹。
11. Urban Rank-Size Rule | 城市位序-规模法则
The rank‑size rule states that in a balanced urban system, the population of a city is inversely proportional to its rank: Pr = P1 / r, where Pr is the population of the r‑th largest city, P1 is the population of the largest city, and r is the rank. A log‑normal city‑size distribution often characterises developed economies, while a primate city dominates in many developing nations.
位序‑规模法则指出，在均衡的城镇体系中，某城市的人口与其位序成反比：Pr = P1 / r。其中 Pr 为第 r 大城市的人口，P1 为首位城市人口，r 为位序。发达经济体常呈现对数正态的城市规模分布，而许多发展中国家则出现首位城市主导的现象。
Deviations from the rank‑size rule signal economic concentration or dispersal. A high primacy index (ratio of first to second city) implies over‑urbanisation in the primate city, often straining infrastructure and fostering regional inequality. In WJEC, you can use this theorem to compare contrasting places and evaluate regeneration policies aiming for polycentric development.
偏离位序‑规模法则意味着经济集中或分散。高首位度指数（首位城市与第二大城市人口之比）预示着首位城市的过度城市化，往往对基础设施造成压力并加剧区域不平等。在 WJEC 课程中，你可以运用这一定理对比不同地区，评价旨在实现多中心发展的再生政策。
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