Lecture 5: Geophysical Fluid Dynamics Review · Geophysical Fluid Dynamics Review Jonathon S....

The equations of motionEffects of spherical geometry and rotation

Balanced flow

Lecture 5:Geophysical Fluid Dynamics Review

Jonathon S. Wright

[email protected]

21 March 2017


Balanced flow

The equations of motionNewtonian mechanics and frame of referenceThe momentum and continuity equationsThermodynamic and concentration equations

Effects of spherical geometry and rotationSpherical geometryThe centrifugal and Coriolis forcesScale analysis

Balanced flowHydrostatic balanceThe primitive equations and the f and β plane approximationsGeostrophic balance and the thermal wind


Balanced flow

Physical law Equation

Conservation of momentum Momentum equations(Navier–Stokes equations)

Conservation of mass Continuity equation

Fluid properties Equation of state

Conservation of salt / water / etc Constituent equations

Conservation of energy Thermodynamic equation


Balanced flow

Newtonian mechanics and frame of referenceThe momentum and continuity equationsThermodynamic and concentration equations

Classical mechanics: two planets

dr1dt

= v1dr2dt

= v2

dv1

dt=−Gm2

(r2 − r1)2r

dv2

dt=

Gm1

(r2 − r1)2r


Balanced flow


ϕ(t)

ϕ(x, y, z, t)

Lagrangian perspective

Eulerian perspective

Fluid motion


Balanced flow


Force per unit mass equals acceleration

Classical mechanics

Newton’s second law: force equals mass times acceleration

F = ma

F

m= a =

dv

dt


Balanced flow


Time rate of change

Advection of momentum

The total derivative

Newton’s second law is valid in the Lagrangian framework. In an Eulerian framework:

d

dtv(x, y, z, t) =

∂v

∂t+∂v

∂x

dx

dt+∂v

∂y

dy

dt+∂v

∂z

dz

dt

=∂v

∂t+ u

∂v

∂x+ v

∂v

∂y+ w

∂v

∂z

=∂v

∂t+ v · ∇v


Balanced flow


u∂ϕ

∂x+ v

∂ϕ

∂y+ w

∂ϕ

∂z= (v · ∇)ϕ

u∂ϕ

∂x

v∂ϕ

∂y

w∂ϕ

∂z

δx

δy

δz

The advection operator

A mathematical expression of the ability of a fluid parcel to carry its properties along


Balanced flow


Forces (per unit mass) acting on the fluid

Conservation of momentum...

The momentum equations

∂v

∂t+ (v · ∇)v =

F

m


Balanced flow


−1

ρ

∂p

∂z

g

p p+ δp

δx

δy

δz



Balanced flow



∂u

∂t+ (v · ∇)u = −1

ρ

∂p

∂x+ Fx

∂v

∂t+ (v · ∇)v = −1

ρ

∂p

∂y+ Fy

∂w

∂t+ (v · ∇)w = −1

ρ

∂p

∂z− g + Fz


Balanced flow


uρ

u+ δuρ+ δρ

δx

δy

δz

Conservation of mass...

convergence or divergence of mass in one dimension

The continuity equation

δyδz

[(ρu)(x, y, z)−

((ρu)(x, y, z) +

∂(ρu)

∂xδx

)]= −∂(ρu)

∂xδxδyδz


Balanced flow


convergence or divergence of mass in three dimensions

time rate of change

The continuity equation

∂ρ

∂tδxδyδz = −

[∂(ρu)

∂x+∂(ρv)

∂y+∂(ρw)

∂z

]δxδyδz

∂ρ

∂t= −

[∂(ρu)

∂x+∂(ρv)

∂y+∂(ρw)

∂z

]

∂ρ

∂t+∇ · (ρv) = 0


Balanced flow


Conservation of momentum and mass...

Four equations

∂u

∂t+ (v · ∇)u = −1

ρ

∂p

∂x+ Fx

∂v

∂t+ (v · ∇)v = −1

ρ

∂p

∂y+ Fy

∂w

∂t+ (v · ∇)w = −1

ρ

∂p

∂z− g + Fz

∂ρ

∂t+∇ · (ρv) = 0


Balanced flow


p = ρRdT

p = ρRdTvvirtual temperature:

Tv = (1 + 0.608q)T

The equation of state

For a dry atmosphere:

For an atmosphere with water vapor (but no clouds):


Balanced flow


ρ = F (T, S, p)

ρ ≈ ρ0

no general formThe equation of state

For the ocean:

For a small volume of ocean (density approximately constant):


Balanced flow


ρ = F (p)

ρ = F (p, · · · )

The equation of state

For a barotropic fluid (density is a function of pressure alone):

For a baroclinic fluid (isopycnals and isobars cross):


Balanced flow


Five equations

∂u

∂t+ (v · ∇)u = −1

ρ

∂p

∂x+ Fx

∂v

∂t+ (v · ∇)v = −1

ρ

∂p

∂y+ Fy

∂w

∂t+ (v · ∇)w = −1

ρ

∂p

∂z− g + Fz

∂ρ

∂t+∇ · (ρv) = 0

ρ = F (p, T, · · · )


Balanced flow


temperature advection (includes adiabatic effects)

time rate of changediabatic heating and cooling

Thermodynamic equation

Conservation of energy

∂θ

∂t+ (v · ∇)θ =

θ

cpTQ


Balanced flow


constituent advection

time rate of change

sources sinks diffusion

Constituent equations

Conservation of water, salt, etc. — we can add as many as we need

∂c

∂t+ (v · ∇)c =

1

ρ(∆csrc −∆csnk + ∆cdiff)


Balanced flow


subject to

boundary

conditions

Seven equations (valid for an inertial frame of reference)

∂u

∂t+ (v · ∇)u = −1

ρ

∂p

∂x+ Fx

∂v

∂t+ (v · ∇)v = −1

ρ

∂p

∂y+ Fy

∂w

∂t+ (v · ∇)w = −1

ρ

∂p

∂z− g + Fz

∂ρ

∂t+∇ · (ρv) = 0

ρ = F (p, T, · · · )∂θ

∂t+ (v · ∇)θ =

θ

cpTQ

∂c

∂t+ (v · ∇)c =

1

ρ(∆csrc −∆csnk + ∆cdiff)


Balanced flow

Spherical geometryThe centrifugal and Coriolis forcesScale analysis

Ω

north pole

equator

south pole

λ

ϑ

r

x

y z

x = r cosϑλ

y = rϑ

z = r − a

u =dx

dt= r cosϑ

dλ

dt

v =dy

dt= r

dϑ

dt

w =dz

dt=dr

dt

Spherical geometry

(x, y, z)→ (λ, ϑ, r)


Balanced flow


Spherical curvature effects in the momentum equations

∂u

∂t+ (v · ∇)u −

(u tanϑ

a

)v +

w

au = −1

ρ

∂p

∂x+ Fx

∂v

∂t+ (v · ∇)v +

(u tanϑ

a

)u+

w

av = −1

ρ

∂p

∂y+ Fy

∂w

∂t+ (v · ∇)w −u

2 + v2

a= −1

ρ

∂p

∂z− g + Fz


Balanced flow


Spherical curvature effects in the continuity equation

∂ρ

∂t+

[∂(ρu)

∂x+

1

cosϑ

∂(ρv cosϑ)

∂y+∂(ρw)

∂z

]= 0

∂ρ

∂t+∇λ,ϑ,r · (ρv) = 0


Balanced flow


... all other equations unchanged except for coordinate and variable substitutions

Spherical curvature effects in the continuity equation

∂ρ

∂t+

[∂(ρu)

∂x+

1

cosϑ

∂(ρv cosϑ)

∂y+∂(ρw)

∂z

]= 0

∂ρ

∂t+∇λ,ϑ,r · (ρv) = 0


Balanced flow


−geff

−ggrv

Fcen = Ω2r⊥

Φ = 0

Ω

ϑ

Effects of rotationThe centrifugal force

Φ = gz +Ω2r2

⊥2

geff = −∇Φ


Balanced flow


Objects leave polestoward Africa

Objects move in straightlines, but Earth rotates

From Earth’s perspective,objects are deflected

Effects of rotationThe Coriolis force Fcor = −2Ω× vR


Balanced flow


Effects of rotationThe Coriolis force Fcor = −2Ω× vR

Ω = (0,Ω cosϑ,Ω, sinϑ) vR = (u, v, w)

Fcor|x = v2Ω sinϑ− w2Ω cosϑ

Fcor|y = −u2Ω sinϑ

Fcor|z = u2Ω cosϑ


Balanced flow


advection PGFcurvature terms

Coriolis terms centrifugal force

friction

Rotational and curvature effects in the momentum equations

∂u

∂t+ (v · ∇)u−

(2Ω sinϑ+

u tanϑ

a

)v +

w

au+ w · 2Ω cosϑ = −1

ρ

∂p

∂x− ∂Φ

∂x+ Fx

∂v

∂t+ (v · ∇)v +

(2Ω sinϑ+

u tanϑ

a

)u+

w

av = −1

ρ

∂p

∂y− ∂Φ

∂y+ Fy

∂w

∂t+ (v · ∇)w − u2 + v2

a− u · 2Ω cosϑ = −1

ρ

∂p

∂z− ∂Φ

∂z+ Fz


Balanced flow


f = 2Ω sinϑ


∂u

∂t+ (v · ∇)u−

(f +

u tanϑ

a

)v +

w


ρ

∂p

∂x− ∂Φ

∂x+ Fx

∂v

∂t+ (v · ∇)v +

(f +

u tanϑ

a

)u+

w

av = −1

ρ

∂p

∂y− ∂Φ

∂y+ Fy

∂w

∂t+ (v · ∇)w − u2 + v2

a− u · 2Ω cosϑ = −1

ρ

∂p

∂z− ∂Φ

∂z+ Fz


Balanced flow


f = 2Ω sinϑ

... all other equations unaffected by rotation


∂u

∂t+ (v · ∇)u−

(f +

u tanϑ

a

)v +

w


ρ

∂p

∂x− ∂Φ

∂x+ Fx

∂v

∂t+ (v · ∇)v +

(f +

u tanϑ

a

)u+

w

av = −1

ρ

∂p

∂y− ∂Φ

∂y+ Fy

∂w

∂t+ (v · ∇)w − u2 + v2

a− u · 2Ω cosϑ = −1

ρ

∂p

∂z− ∂Φ

∂z+ Fz


Balanced flow


u, v ∼ U

x, y ∼ L

1

ρ∇pxy ∼ PGFxy

f ∼ f0

w ∼ W

z ∼ H

1

ρ∇pz ∼ PGFz

time ∼ T =L

U

Scale analysis

Use typical scales of motion for large-scale dynamics to simplify the equations


Balanced flow


10−4 10−4 10−3 10−5 10−8 10−6 10−3 ?

10−7 10−7 10−5 10−3 10 10 ?

Scale analysis: mid-latitude atmospheric weather systems

∂u

∂t+ (v · ∇)u−

(2Ω sinϑ+

u tanϑ

a

)v +

w


ρ

∂p

∂x+ Fx

∂v

∂t+ (v · ∇)v +

(2Ω sinϑ+

u tanϑ

a

)u+

w

av = −1

ρ

∂p

∂y+ Fy

∂w

∂t+ (v · ∇)w − u2 + v2

a− u · 2Ω cosϑ = −1

ρ

∂p

∂z− g + Fz


Balanced flow

Hydrostatic balanceThe primitive equations and the f and β plane approximationsGeostrophic balance and the thermal wind

10−4 10−4 10−3 10−5 10−8 10−6 10−3 ?

10−7 10−7 10−5 10−3 10 10 ?

Hydrostatic balance

∂u

∂t+ (v · ∇)u−

(2Ω sinϑ+

u tanϑ

a

)v +

w


ρ

∂p

∂x+ Fx

∂v

∂t+ (v · ∇)v +

(2Ω sinϑ+

u tanϑ

a

)u+

w

av = −1

ρ

∂p

∂y+ Fy

∂w

∂t+ (v · ∇)w − u2 + v2

a− u · 2Ω cosϑ = −1

ρ

∂p

∂z− g + Fz


Balanced flow


Hydrostatic balance

∂p

∂z= −ρg

valid for

(H

L

)2

1


Balanced flow


Applying hydrostatic balance: constant pressure coordinates

∂Φ

∂p=∂Φ

∂z

∂z

∂p= − 1

ρg

∂Φ

∂z= −1

ρ= −RdT

p

∇pp = 0⇒ ∇zp +∂p

∂z∇pz = ∇zp− ρg∇pz = 0

1

ρ∇zp = g∇pz = ∇pΦ


Balanced flow


this perspective simplifies the equations, but complicates the lower boundary conditions

Applying hydrostatic balance: constant pressure coordinates

∂u

∂t+ (u · ∇p)u+ ω

∂u

∂p− fv = −∂Φ

∂x+ Fx

∂v

∂t+ (u · ∇p)v + ω

∂v

∂p+ fu = −∂Φ

∂y+ Fy

∂Φ

∂p= −RdT

p= −1

ρ

∇p · u +∂ω

∂p= 0

Q = cpdT

dt− 1

ρ

dp

dt= cp

(∂T

∂t+ (u · ∇p)T + ω

∂T

∂p

)− 1

ρω


Balanced flow


10−4 10−4 10−3 10−5 10−8 10−6 10−3 ?

10−7 10−7 10−5 10−3 10 10 ?

The primitive equations

∂u

∂t+ (v · ∇)u−

(2Ω sinϑ+

u tanϑ

a

)v +

w


ρ

∂p

∂x+ Fx

∂v

∂t+ (v · ∇)v +

(2Ω sinϑ+

u tanϑ

a

)u+

w

av = −1

ρ

∂p

∂y+ Fy

∂w

∂t+ (v · ∇)w − u2 + v2

a− u · 2Ω cosϑ = −1

ρ

∂p

∂z− g + Fz


Balanced flow


10−4 10−4 10−3 10−5 10−8 10−6 10−3 ?

10−7 10−7 10−5 10−3 10 10 ?

The f -plane

∂u

∂t+ (v · ∇)u−

(2Ω sinϑ+

u tanϑ

a

)v +

w


ρ

∂p

∂x+ Fx

∂v

∂t+ (v · ∇)v +

(2Ω sinϑ+

u tanϑ

a

)u+

w

av = −1

ρ

∂p

∂y+ Fy

∂w

∂t+ (v · ∇)w − u2 + v2

a− u · 2Ω cosϑ = −1

ρ

∂p

∂z− g + Fz


Balanced flow


∂u

∂t+ (v · ∇)u− f0v = −1

ρ

∂p

∂x+ Fx

∂v

∂t+ (v · ∇)v + f0u = −1

ρ

∂p

∂y+ Fy

∂w

∂t+ (v · ∇)w = −1

ρ

∂p

∂z+ Fz

focus on a smallpart of the sphere

. f = f0 is assumedconstant

. curvature can beneglected

The f -plane


Balanced flow


∂u

∂t+ (v · ∇)u− fv = −1

ρ

∂p

∂x+ Fx

∂v

∂t+ (v · ∇)v + fu = −1

ρ

∂p

∂y+ Fy

∂w

∂t+ (v · ∇)w = −1

ρ

∂p

∂z+ Fz

focus on a smallpart of the sphere

. f ≈ f0 + βy

. curvature can beneglected

Retains the simpler geometry ofthe plane while allowing f to vary

by linearizing f : f = f0 + βy

The β-plane


Balanced flow


10−4 10−4 10−3 10−5 10−8 10−6 10−3 ?

keep only the largest terms in the horizontal momentum equations:

fug = −1

ρ

∂p

∂y−fvg = −f

ρ

∂p

∂x

Geostrophic balance

∂u

∂t+ (v · ∇)u−

(2Ω sinϑ+

u tanϑ

a

)v +

w


ρ

∂p

∂x+ Fx

∂v

∂t+ (v · ∇)v +

(2Ω sinϑ+

u tanϑ

a

)u+

w

av = −1

ρ

∂p

∂y+ Fy


Balanced flow


fug = −1

ρ

∂p

∂y−fvg = −f

ρ

∂p

∂x

Valid for U/L f0

Rossby number: Ro ≡ U

f0L

Geostrophic balance


Balanced flow


vg

pressuregradient force

Coriolisforce

Low pressure

High pressure

Geostrophic balance


Balanced flow


2.8 Geostrophic and Thermal Wind Balance 87

Fig. 2.5 Schematic of geostrophic flow with a positive value of the Coriolisparameter f . Flow is parallel to the lines of constant pressure (isobars). Cy-clonic flow is anticlockwise around a low pressure region and anticyclonic flowis clockwise around a high. If f were negative, as in the Southern hemisphere,(anti-)cyclonic flow would be (anti-)clockwise.

? If the Coriolis force is constant and if the density does not vary in the horizontalthe geostrophic flow is horizontally non-divergent and

rz · ug =@ug@x

+ @vg@y

= 0 . (2.189)

We may define the geostrophic streamfunction, , by

pf00

, (2.190)

whence

ug = @ @y

, vg =@ @x

. (2.191)

The vertical component of vorticity, , is then given by

= k ·r v = @v@x

@u@y

= r2z . (2.192)

? If the Coriolis parameter is not constant, then cross-differentiating (2.187) gives,for constant density geostrophic flow,

vg@f@y

+ frz · ug = 0, (2.193)

which implies, using mass continuity,

vg = f@w@z

. (2.194)

anticyclone

cyclone

from Vallis 2006

Geostrophic balance

http://vallisbook.org


Balanced flow


10−4 10−4 10−3 10−5 10−8 10−6 10−3 ?

Friction slows the near-surface wind, sothat geostrophic balance is not exact

Quasi-geostrophic balance

∂u

∂t+ (v · ∇)u−

(2Ω sinϑ+

u tanϑ

a

)v +

w


ρ

∂p

∂x+ Fx

∂v

∂t+ (v · ∇)v +

(2Ω sinϑ+

u tanϑ

a

)u+

w

av = −1

ρ

∂p

∂y+ Fy


Balanced flow


vgv

friction

pressuregradient force

Coriolisforce

Low pressure

High pressure



Balanced flow


2.8 Geostrophic and Thermal Wind Balance 87

Fig. 2.5 Schematic of geostrophic flow with a positive value of the Coriolisparameter f . Flow is parallel to the lines of constant pressure (isobars). Cy-clonic flow is anticlockwise around a low pressure region and anticyclonic flowis clockwise around a high. If f were negative, as in the Southern hemisphere,(anti-)cyclonic flow would be (anti-)clockwise.

? If the Coriolis force is constant and if the density does not vary in the horizontalthe geostrophic flow is horizontally non-divergent and

rz · ug =@ug@x

+ @vg@y

= 0 . (2.189)

We may define the geostrophic streamfunction, , by

pf00

, (2.190)

whence

ug = @ @y

, vg =@ @x

. (2.191)

The vertical component of vorticity, , is then given by

= k ·r v = @v@x

@u@y

= r2z . (2.192)

? If the Coriolis parameter is not constant, then cross-differentiating (2.187) gives,for constant density geostrophic flow,

vg@f@y

+ frz · ug = 0, (2.193)

which implies, using mass continuity,

vg = f@w@z

. (2.194)

divergentclear and dry

convergentcloudy and wet

from Vallis 2006


http://vallisbook.org


Balanced flow


Assuming geostrophic

(fug = −∂Φ

∂y

∣∣∣∣p

)and hydrostatic

(∂Φ

∂p= −RdT

p

)balance:

f∂ug

∂p= − ∂

∂p

∂Φ

∂y

∣∣∣∣p

=Rd

p

∂T

∂y

∣∣∣∣p

f∂vg

∂p=Rd

pz×∇pT

The vertical gradient of the geostrophic wind dependson the horizontal gradient of temperature

The thermal wind balance


Balanced flow


−∇p

−∇p

·

×

u > 0

u < 0

Higher pressure Lower pressure

Lower pressure Higher pressure

WarmLight

ColdDense

tropics trade winds subtropics ϑ

The thermal wind balance


Balanced flow


Assuming geostrophic

(fug = − 1

ρ0

∂p

∂y

)and hydrostatic

(∂p

∂z= −ρg

)balance:

f∂ug

∂z= − 1

ρ0

∂

∂z

∂p

∂y= − 1

ρ0

∂

∂y

∂p

∂z=

g

ρ0

∂ρ

∂y

f∂vg

∂z= − g

ρ0z×∇ρ

The vertical gradient of the geostrophic current dependson the horizontal gradient of density

Thermal “wind” in the ocean

Lecture 5: Geophysical Fluid Dynamics Review · Geophysical Fluid Dynamics Review Jonathon S....

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