3-D, rotating flow, 3-D conservation laws. ρ_{t} + (ρu)_{x} + (ρv)_{y} + (ρw)_{z} = 0, (ρz)_{t} + (ρuz)_{x} + (ρvz)_{y} + (ρwz)_{z} = ρw,

(ρu)_{t} + (p + ρuu)_{x} + (ρvu)_{y} + (ρwu)_{z} − ρfv = 0, (ρv)_{t} + (ρuv)_{x} + (p + ρvv)_{y} + (ρwv)_{z} + ρfu = 0, (ρw)_{t} + (ρuw)_{x} + (ρvw)_{y} + (p + ρww)_{z} + ρg = 0,

(ρe)_{t} + ∇ · ((p + ρe)u) = ∇ · (k∇ϑ) + ρq_{r}.

3-D, rotating flow, 2-D conservation laws, mean values.

μ_{t} + (μU)_{x} + (μV)_{y} = 0, (exact) (μZ)_{t} + (μUZ)_{x} + (μV Z)_{y} = μW − ⟨ρũz̃⟩_{x} − ⟨ρṽz̃⟩_{y}, (μU)_{t} + (⟨p⟩ + μUU)_{x} + (μV U)_{y} − μfV + p_{B}B_{x} = 0, (μV)_{t} + (μUV)_{x} + (⟨p⟩ + μV V)_{y} + μfU + p_{B}B_{y} = 0,

(μW)_{t} + (μUW)_{x} + (μV W)_{y} + μg = p_{B}, (μE)_{t} + (U(⟨p⟩ + μE))_{x} + (V (⟨p⟩ + μE))_{y} = μQ_{r} + p_{B}B_{t}.

\frac{Dμ}{Dt} + μ(U_{x} + V_{y}) = 0,

\frac{DU}{Dt} − fV + \frac{1}{μ}⟨p⟩_{x} + GB_{x} = 0,

\frac{DV}{Dt} + fU + \frac{1}{μ}⟨p⟩_{y} + GB_{y} = 0,

\frac{DZ}{Dt} = W, \frac{DW}{Dt} + g = G,

\frac{DE}{Dt} + \frac{1}{μ} ((U⟨p⟩)_{x} + (V ⟨p⟩)_{y}) = Q_{r} + GB_{t}.

The MD, shallow atmosphere approximation

⟨p⟩ = μRT = μG(Z − B) = Aμ^{α}G^{δ}

\frac{Dμ}{Dt} + μ(U_{x} + V_{y}) = 0 and \frac{1}{A} \frac{DA}{Dt} = δ \frac{Q_{r}}{RT} ,

\frac{DU}{Dt} − fV + \frac{1}{μ}⟨p⟩_{x} + GB_{x} = 0 and \frac{DV}{Dt} + fU + \frac{1}{μ}⟨p⟩_{y} + GB_{y} = 0,

\frac{DW}{Dt} + g = G and \frac{1}{ω^{2}} \frac{DG}{Dt} + W = Ḃ + \frac{δ}{G} (Q_{r} − RT (U_{x} + V_{y}))