1. Conservation of Momentum (horizontal)
𝑑 𝑉 𝑑𝑡 =− 1 𝜌 ∇ 𝑝 −𝑓 𝑘 × 𝑉 + 𝑚
Conservation of Momentum (vertical)
𝜕𝑝 𝜕𝑧 = −𝜌𝑔
Conservation of Mass
𝜕𝜌 𝜕𝑡 = −∇∙ 𝜌 𝑉 − 𝜕 𝜕𝑧 (𝜌𝑤)
First Law of Thermodynamics (Cons. of Energy)
𝜕 𝜕𝑡 ln 𝜃 = − 𝑉 ∙∇ ln 𝜃 −𝑤 𝜕 𝜕𝑧 ln 𝜃 − 𝑄 𝐶 𝑝 𝑇

2. 4 Equations, but 6 variables: 𝑉 ,w, p, 𝜌, 𝑇, 𝜃
Close the system with 2 additional relationships: Ideal gas law: 𝑝= 𝜌𝑅𝑇 Definition of potential temperature: θ= 𝑝 𝑜 𝑝 𝜅 T Here, R, po and 𝜅 are specified constants, 𝐹 is the frictional force and Q is diabatic heating.