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The equations of motionEffects of spherical geometry and rotation
Balanced flow
Lecture 5:Geophysical Fluid Dynamics Review
Jonathon S. Wright
jswright@tsinghua.edu.cn
21 March 2017
The equations of motionEffects of spherical geometry and rotation
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
The equations of motionEffects of spherical geometry and rotation
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
The equations of motionEffects of spherical geometry and rotation
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
The equations of motionEffects of spherical geometry and rotation
Balanced flow
Newtonian mechanics and frame of referenceThe momentum and continuity equationsThermodynamic and concentration equations
ϕ(t)
ϕ(x, y, z, t)
Lagrangian perspective
Eulerian perspective
Fluid motion
The equations of motionEffects of spherical geometry and rotation
Balanced flow
Newtonian mechanics and frame of referenceThe momentum and continuity equationsThermodynamic and concentration equations
Force per unit mass equals acceleration
Classical mechanics
Newton’s second law: force equals mass times acceleration
F = ma
F
m= a =
dv
dt
The equations of motionEffects of spherical geometry and rotation
Balanced flow
Newtonian mechanics and frame of referenceThe momentum and continuity equationsThermodynamic and concentration equations
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
The equations of motionEffects of spherical geometry and rotation
Balanced flow
Newtonian mechanics and frame of referenceThe momentum and continuity equationsThermodynamic and concentration equations
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
The equations of motionEffects of spherical geometry and rotation
Balanced flow
Newtonian mechanics and frame of referenceThe momentum and continuity equationsThermodynamic and concentration equations
Forces (per unit mass) acting on the fluid
Conservation of momentum...
The momentum equations
∂v
∂t+ (v · ∇)v =
F
m
The equations of motionEffects of spherical geometry and rotation
Balanced flow
Newtonian mechanics and frame of referenceThe momentum and continuity equationsThermodynamic and concentration equations
−1
ρ
∂p
∂z
g
p p+ δp
δx
δy
δz
The momentum equations
The equations of motionEffects of spherical geometry and rotation
Balanced flow
Newtonian mechanics and frame of referenceThe momentum and continuity equationsThermodynamic and concentration equations
The momentum 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
The equations of motionEffects of spherical geometry and rotation
Balanced flow
Newtonian mechanics and frame of referenceThe momentum and continuity equationsThermodynamic and concentration equations
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
The equations of motionEffects of spherical geometry and rotation
Balanced flow
Newtonian mechanics and frame of referenceThe momentum and continuity equationsThermodynamic and concentration equations
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
The equations of motionEffects of spherical geometry and rotation
Balanced flow
Newtonian mechanics and frame of referenceThe momentum and continuity equationsThermodynamic and concentration equations
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
The equations of motionEffects of spherical geometry and rotation
Balanced flow
Newtonian mechanics and frame of referenceThe momentum and continuity equationsThermodynamic and concentration equations
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):
The equations of motionEffects of spherical geometry and rotation
Balanced flow
Newtonian mechanics and frame of referenceThe momentum and continuity equationsThermodynamic and concentration equations
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):
The equations of motionEffects of spherical geometry and rotation
Balanced flow
Newtonian mechanics and frame of referenceThe momentum and continuity equationsThermodynamic and concentration equations
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):
The equations of motionEffects of spherical geometry and rotation
Balanced flow
Newtonian mechanics and frame of referenceThe momentum and continuity equationsThermodynamic and concentration equations
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, · · · )
The equations of motionEffects of spherical geometry and rotation
Balanced flow
Newtonian mechanics and frame of referenceThe momentum and continuity equationsThermodynamic and concentration equations
temperature advection (includes adiabatic effects)
time rate of changediabatic heating and cooling
Thermodynamic equation
Conservation of energy
∂θ
∂t+ (v · ∇)θ =
θ
cpTQ
The equations of motionEffects of spherical geometry and rotation
Balanced flow
Newtonian mechanics and frame of referenceThe momentum and continuity equationsThermodynamic and concentration equations
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)
The equations of motionEffects of spherical geometry and rotation
Balanced flow
Newtonian mechanics and frame of referenceThe momentum and continuity equationsThermodynamic and concentration equations
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)
The equations of motionEffects of spherical geometry and rotation
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)
The equations of motionEffects of spherical geometry and rotation
Balanced flow
Spherical geometryThe centrifugal and Coriolis forcesScale analysis
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
The equations of motionEffects of spherical geometry and rotation
Balanced flow
Spherical geometryThe centrifugal and Coriolis forcesScale analysis
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
The equations of motionEffects of spherical geometry and rotation
Balanced flow
Spherical geometryThe centrifugal and Coriolis forcesScale analysis
... 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
The equations of motionEffects of spherical geometry and rotation
Balanced flow
Spherical geometryThe centrifugal and Coriolis forcesScale analysis
−geff
−ggrv
Fcen = Ω2r⊥
Φ = 0
Ω
ϑ
Effects of rotationThe centrifugal force
Φ = gz +Ω2r2
⊥2
geff = −∇Φ
The equations of motionEffects of spherical geometry and rotation
Balanced flow
Spherical geometryThe centrifugal and Coriolis forcesScale analysis
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
The equations of motionEffects of spherical geometry and rotation
Balanced flow
Spherical geometryThe centrifugal and Coriolis forcesScale analysis
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
The equations of motionEffects of spherical geometry and rotation
Balanced flow
Spherical geometryThe centrifugal and Coriolis forcesScale analysis
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
The equations of motionEffects of spherical geometry and rotation
Balanced flow
Spherical geometryThe centrifugal and Coriolis forcesScale analysis
f = 2Ω sinϑ
Rotational and curvature effects in the momentum equations
∂u
∂t+ (v · ∇)u−
(f +
u tanϑ
a
)v +
w
au+ w · 2Ω cosϑ = −1
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
The equations of motionEffects of spherical geometry and rotation
Balanced flow
Spherical geometryThe centrifugal and Coriolis forcesScale analysis
f = 2Ω sinϑ
... all other equations unaffected by rotation
Rotational and curvature effects in the momentum equations
∂u
∂t+ (v · ∇)u−
(f +
u tanϑ
a
)v +
w
au+ w · 2Ω cosϑ = −1
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
The equations of motionEffects of spherical geometry and rotation
Balanced flow
Spherical geometryThe centrifugal and Coriolis forcesScale analysis
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
The equations of motionEffects of spherical geometry and rotation
Balanced flow
Spherical geometryThe centrifugal and Coriolis forcesScale analysis
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
au+ w · 2Ω cosϑ = −1
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
The equations of motionEffects of spherical geometry and rotation
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
au+ w · 2Ω cosϑ = −1
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
The equations of motionEffects of spherical geometry and rotation
Balanced flow
Hydrostatic balanceThe primitive equations and the f and β plane approximationsGeostrophic balance and the thermal wind
Hydrostatic balance
∂p
∂z= −ρg
valid for
(H
L
)2
1
The equations of motionEffects of spherical geometry and rotation
Balanced flow
Hydrostatic balanceThe primitive equations and the f and β plane approximationsGeostrophic balance and the thermal wind
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
The equations of motionEffects of spherical geometry and rotation
Balanced flow
Hydrostatic balanceThe primitive equations and the f and β plane approximationsGeostrophic balance and the thermal wind
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
The equations of motionEffects of spherical geometry and rotation
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 ?
The primitive equations
∂u
∂t+ (v · ∇)u−
(2Ω sinϑ+
u tanϑ
a
)v +
w
au+ w · 2Ω cosϑ = −1
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
The equations of motionEffects of spherical geometry and rotation
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 ?
The f -plane
∂u
∂t+ (v · ∇)u−
(2Ω sinϑ+
u tanϑ
a
)v +
w
au+ w · 2Ω cosϑ = −1
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
The equations of motionEffects of spherical geometry and rotation
Balanced flow
Hydrostatic balanceThe primitive equations and the f and β plane approximationsGeostrophic balance and the thermal wind
∂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
The equations of motionEffects of spherical geometry and rotation
Balanced flow
Hydrostatic balanceThe primitive equations and the f and β plane approximationsGeostrophic balance and the thermal wind
∂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
The equations of motionEffects of spherical geometry and rotation
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 ?
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
au+ w · 2Ω cosϑ = −1
ρ
∂p
∂x+ Fx
∂v
∂t+ (v · ∇)v +
(2Ω sinϑ+
u tanϑ
a
)u+
w
av = −1
ρ
∂p
∂y+ Fy
The equations of motionEffects of spherical geometry and rotation
Balanced flow
Hydrostatic balanceThe primitive equations and the f and β plane approximationsGeostrophic balance and the thermal wind
fug = −1
ρ
∂p
∂y−fvg = −f
ρ
∂p
∂x
Valid for U/L f0
Rossby number: Ro ≡ U
f0L
Geostrophic balance
The equations of motionEffects of spherical geometry and rotation
Balanced flow
Hydrostatic balanceThe primitive equations and the f and β plane approximationsGeostrophic balance and the thermal wind
vg
pressuregradient force
Coriolisforce
Low pressure
High pressure
Geostrophic balance
The equations of motionEffects of spherical geometry and rotation
Balanced flow
Hydrostatic balanceThe primitive equations and the f and β plane approximationsGeostrophic balance and the thermal wind
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
The equations of motionEffects of spherical geometry and rotation
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 ?
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
au+ w · 2Ω cosϑ = −1
ρ
∂p
∂x+ Fx
∂v
∂t+ (v · ∇)v +
(2Ω sinϑ+
u tanϑ
a
)u+
w
av = −1
ρ
∂p
∂y+ Fy
The equations of motionEffects of spherical geometry and rotation
Balanced flow
Hydrostatic balanceThe primitive equations and the f and β plane approximationsGeostrophic balance and the thermal wind
vgv
friction
pressuregradient force
Coriolisforce
Low pressure
High pressure
Quasi-geostrophic balance
The equations of motionEffects of spherical geometry and rotation
Balanced flow
Hydrostatic balanceThe primitive equations and the f and β plane approximationsGeostrophic balance and the thermal wind
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
Quasi-geostrophic balance
The equations of motionEffects of spherical geometry and rotation
Balanced flow
Hydrostatic balanceThe primitive equations and the f and β plane approximationsGeostrophic balance and the thermal wind
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
The equations of motionEffects of spherical geometry and rotation
Balanced flow
Hydrostatic balanceThe primitive equations and the f and β plane approximationsGeostrophic balance and the thermal wind
−∇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
The equations of motionEffects of spherical geometry and rotation
Balanced flow
Hydrostatic balanceThe primitive equations and the f and β plane approximationsGeostrophic balance and the thermal wind
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
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