PDF Forces - atmdyn.org

1. Pressure gradient force 2. Gravitational force

3. Viscous force

Forces

Pressure

gradient Coriolis

Pressure gradient force (PGF)

A fluid parcel placed in a pressure gradient is subjected to a net force associated with pressure difference between one side and the other.

* Recall pressure is force per a unit surface, [N/m²] 


(by molecules that bounce off the surface, randomly)

(from environment Canada, ec.gc.ca)

air parcel

Pressure gradient force (PGF)

pA

pB

A fluid parcel placed in a pressure gradient is subjected to a net force associated with pressure difference between one side and the other.

* Recall pressure is force per a unit surface, [N/m²] 


(by molecules that bounce off the surface, randomly)

Pressure gradient force (PGF)

δz

δxm δy

F_Ax = −p_Aδ^yδ^z

F!"

PGF

m = −∇p vector form

m = ρδxδyδz

F_x

m = − (p_A − p_B)δ^yδ^z ρδ^xδ^yδ^z

pA

pB

F_Bx = p_Bδ^yδ^z

F_x

m = − 1 ρ

∂p

∂x

F_y

m = − 1 ρ

∂p

∂y

F_z

m = − 1 ρ

∂p

∂z

A fluid parcel placed in a pressure gradient is subjected to a net force associated with pressure difference between one side and the other.

Newton’s law of universal gravitation

Any two elements of mass attract each other with a force

proportional to their masses and inversely proportional to the square of distance.

Gravitational force

M

m

F!" r

G = − GMm r² r! F!"

G

m = − GM

r² r! ≈ − GM

a² r! = g^*₀

Internal friction in a fluid. Viscosity arises when the fluid velocity varies spatially so that random molecular (or small-scale eddy)

motion makes a net transport of momentum from faster-moving parcel to slower-moving one… (Intro. dyn. Met., Holton).

Viscous force

Internal friction in a fluid. Viscosity arises when the fluid velocity varies spatially so that random molecular (or small-scale eddy)

motion makes a net transport of momentum from faster-moving parcel to slower-moving one… (Intro. dyn. Met., Holton).

Viscous force

F_rx

m = ν ^∂

2u

∂z²

δz

δx δy

m

m = ρδxδyδz (τzx)A

(τzx)B

u (m/s) z (m)

τ ≈ µ ^∂u

∂z

(ν = µ ^/ ρ⁾ assuming μ = const

F_rx

m = (τ _A ⁻τ _B⁾δ^xδ^y ρδ^xδ^yδ^z F_rx

m = 1 ρ

∂τ

∂z F_rx

m = 1 ρ

∂

∂z µ ^∂u

∂z

"

#$ %

&

'

Viscous force

δz

δx δy

m

m = ρδxδyδz (τzx)A

(τzx)B

u (m/s) z (m)

F_rx

m = (τ _A ⁻τ _B⁾δ^xδ^y ρδ^xδ^yδ^z F_rx

m = 1 ρ

∂τ

∂z τ ≈ µ ^∂u

∂z F_rx

m = 1 ρ

∂

∂z µ ^∂u

∂z

"

#$ %

&

'

F_rx

m = ν ^∂

2u

∂x² + ∂²u

∂y² + ∂²u

∂z²

"

#$ %

&

' = ν∇²u (ν = µ ^/ ρ⁾

generalized 
 form in x-dir

assuming μ = const

Internal friction in a fluid. Viscosity arises when the fluid velocity varies spatially so that random molecular (or small-scale eddy)

motion makes a net transport of momentum from faster-moving parcel to slower-moving one… (Intro. dyn. Met., Holton).

Viscous force

δz

δx δy

m

m = ρδxδyδz (τzx)A

(τzx)B

u (m/s) z (m)

τ ≈ µ ^∂u

∂z F_rx

m = ν ^∂

2u

∂x² + ∂²u

∂y² + ∂²u

∂z²

"

#$ %

&

' =ν∇²u

(ν = µ ^/ ρ⁾ In all direction

Internal friction in a fluid. Viscosity arises when the fluid velocity varies spatially so that random molecular (or small-scale eddy)

motion makes a net transport of momentum from faster-moving parcel to slower-moving one… (Intro. dyn. Met., Holton).

F_ry

m = ν ^∂

2v

∂x² + ∂²v

∂y² + ∂²v

∂z²

"

#$ %

&

' = ν∇²v

F_rz

m = ν ^∂

2w

∂x² + ∂²w

∂y² + ∂² w

∂z²

"

#$ %

&

' = ν∇²w

F!"

r

m = ν∇²u!

Loss of momentum from maximum value to surrounding region (diffuse out momentum…)


Viscous force

du

dt = ν ^∂

2u

∂z² F_rx

m = ν ^∂

2u

∂z²

D! v Dt =

F!₁ m +

F!₂

m +"

u

∂

²

u/∂z

²

t=0 t=1

t=∞

du

dt = ν ^∂

2u

∂z² ∝ −ν^u

z