PDF Background Concepts

Atmospheric dynamics 1

Background Concepts

Variables, laws

• Basic variables: 6 (7)


• Need 6 (7) physical laws 


(3) conservation of momentum





(1) conservation of energy





(1) equation of state





(1) conservation of mass





(1) conservation of mass for composition

u, v, w, p, ρ, T, (q, S)

Concepts

• Continuum hypothesis


- parcel: tiny volume >> such that variables are well defined 
 tiny volume << length scale of variables are varying

• Material derivative 
 


T = T(x, y, z, t)

DT

Dt ≡ ∂T

∂t + u ∂T

∂x + v ∂T

∂y + w ∂T

∂z

Equations

• Momentum equation 
 F = ma





Pressure gradient force





Viscosity

• Thermodynamic energy equation 















I = I(α, η, s)

dI = dQ + dW + dC

c_p DT

Dt − α Dp

Dt = J

Equations

• Conservation of mass (continuity equation)











• Conservation of mass for composition 
 


∂

∂t ∫_V ρ dv = − ∫_S ρ v ⃗ ⋅ ⃗s = − ∫_V ∇ ⋅ (ρ v )⃗ dv

∂ρ

∂t + ∇ ⋅ (ρ v ⃗) = 0

∂

∂t ∫_V ρq dv = − ∫_S ρq v ⃗ ⋅ ⃗s = − ∫_V ∇ ⋅ (ρq v ⃗) dv

∂q

∂t + ∇ ⋅ (q v ) = 0⃗

Equations

• Conservation of mass for composition

• If q-flux include “downgrading mixing”

∂q

∂t + ∇ ⋅ (q v ) = 0⃗

∂

∂t ∫_V ρq dv = − ∫_S (ρq v ⃗ − ρκ∇q) ⋅ ⃗s

∂q

∂t + ∇ ⋅ (q v ) =⃗ κ∇²q