Chapter 3
Physical Processes in the Climate System
3.1 Conservation of Momentum
3.2 Equation of State
3.3 Temperature Equation
3.4 Continuity Equation
3.5 Moisture and Salinity Equation
3.6 Moist Processes
3.7 Wave Processes in the Atmosphere and Ocean
3.8 Overview (equals shortcut/level of “need to know”)
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
Includes equations that are budgets for these conservation
laws, and discussion of main balances.
Why:
• Climate models are based on these equations and relationships
• We can use the main balances to understand features of
current climate, El Niño, global warming,..
Know:
• What we do with the balances & concepts (not equation
details)
• Example (from Section 3.8 of text): Preview of “Section 3.1 Overview”
• An approximate balance between the Coriolis force and the
pressure gradient force holds for winds and currents in many
applications (geostrophic balance) (Fig. 3.4).
• Will come in handy for El Niño …
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
3.1 Conservation of Momentum
Newton:
ma = F (Eq. 3.1)
F - Force
m - Mass
a - Acceleration
F
a= m
Use force per unit mass for atm/oc.
•Acceleration a = rate of change of velocity
d velocity = Coriolis+PGF+gravity+F
drag
dt
eqs. 3.4 & 3.5
• Coriolis force: due to rotation of earth (apparent force).
• PGF: pressure gradient force. Tends to move air from high
to low pressure.
• Fdrag: friction-like forces due to turbulent or surface drag.
• gravity g = 9.8 m/s2.
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
Figure 3.1
Schematic of directions and velocities
Coordinate system
for directions and
velocities. Blown up
region shows local
Cartesian coordinate
system (for each
region of the sphere).
Distances east, north
& up are x, y, z.
(Also lat, lon j, l)
Velocity components
u, v, w (eastward,
northward, up).
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
Figure 3.2
Schematic of the Coriolis force
• Apparent force that acts on moving masses in a rotating
reference frame (Earth rotates once per day)
• a leading effect in atm/ocean motions (time scales > day, little friction)
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
Coriolis force animation
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
Coriolis force (cont.)
• Turns a body or air/water parcel to the right in the northern
hemisphere; to the left in the southern hemisphere.
• Exactly on the equator, the horizontal component of the
Coriolis force is zero
• Acts only for bodies moving relative to the surface of the
Earth’s equator and is proportional to velocity.
• Constant of proportionality is f =(4/1day)sin(latitude), known
as the Coriolis parameter; f is positive in the northern
hemisphere, negative in the southern hemisphere, and zero at
the equator.
• [northward force -fu (to right of eastward wind component).]
• [eastward force fv (to right of northward wind component).]
• [vertical component of Coriolis <<gravity so less important]
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
Coriolis force (cont.)
“Beta effect” change of Coriolis force with latitude
df

dy
•  is proportional to cos(latitude)
• Always positive and maximum at the equator
• matters because Coriolis so important
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
Figure 3.3
Pressure Gradient Force: e.g., pressure p decreasing eastward
p
x
[Pressure: force per unit area. Change dp across distance dx
gives force per volume. High to low, so – dp/ dx .
Divide by density r (mass per volume) to get force per mass.]
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
Pressure Gradient Force, PGF
• Force per unit mass, tends to accelerate air from higher
to lower pressure
_ 1 p
• x direction
r x
_ 1 p
• y direction
r y
p
x
• partial derivatives: pressure change eastward and northward,
respectively
• density r (mass per unit volume)
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
Horizontal momentum equations
(Horizontal = perpendicular to gravity)
d velocity = Coriolis+PGF+gravity+F
Eq. 3.2
drag
dt
du
1 p
_
x
Eq. 3.4
x-direction (east)
= fv
+
F
drag
dt
r x
y-direction (north)
dv
_ _ 1 p + F y
Eq. 3.5
= fu
drag
dt
r y
Largest terms
(usually)
Note: PGF depends on what’s happening to either side;
neighboring regions affect each other
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
Figure 3.4
Schematic of geostrophic wind and wind with frictional effects
Geostrophic balance:
At large scales at mid-latitudes and
approaching the tropics the Coriolis
force and the pressure gradient force
are the dominant forces
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
_ 1 p
fug =
r y
1 p
fvg =
r x
Pressure-height relation: Hydrostatic balance
dp _
= rg
dz
Eq. 3.8
• Comes from vertical momentum equation but
dominated* by balance between vertical pressure gradient
and gravity
• Pressure at each level in atmosphere or ocean is given by
g times amount of mass above it
[p=zrg dz]
*except in small scale features (thunderstorms, squall lines, etc.)
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
Section 3.1 Overview
• An approximate balance between the Coriolis force and the
pressure gradient force holds for winds and currents in many
applications (geostrophic balance) (Fig. 3.4).
• The Coriolis force tends to turn a flow to the right of its
motion in the Northern Hemisphere (left in the Southern
Hemisphere); the pressure gradient force acts from high toward
low pressure.
• The Coriolis parameter f varies with latitude (zero at the
equator, increasing to the north, negative to the south); this is
called the beta-effect ( = rate of change of f with latitude).
• In the vertical direction, the pressure gradient force balances
gravity (hydrostatic balance). This allows us to use pressure as a
vertical coordinate. Pressure is proportional to the mass above
in the atmospheric or oceanic column.
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
3.2 Equation of State
• Relates density to temperature T, pressure p (+other factors)
• For the atmosphere: Ideal gas law
p
r=
Eq. 3.10
RT
• density r decreases with temperature, incr. with pressure
• Ideal gas constant R = 287 J kg-1 K-1, T in Kelvin
• For the ocean:
• density an empirical function of T, p, and Salinity S
• For small T changes can use coeff. of thermal expansion eT
(percent density decrease per C of T increase), i.e.
r-r0= -r0 e T(T-T0),
with eT =2.710-4C-1 for refc temp T0=22C
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
Figure 3.5
Application: thermal circulation
UCLA (sea breeze)
SM Bay
Tropics
e.g.: (Hadley circ)
subtropics
West Pacific
(Walker circ.)
East Pacific
• relatively low pressure (at given height) at low levels in
warm region; PGF toward warm region (near surface)
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
• hydrostatic + ideal gas law gives:
• fixed mass per area between pressure surfaces
• Warm air less dense
greater column thickness (height
difference) between two pressure surfaces
• p1 below z1 in warm region, so p at z1 is L rel to cool region
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
Application: sea level rise by thermal expansion
• mass of ocean: area*rh. If mass & area constant, while
density decreases by dr, depth h must change by dh
hdr = -rdh
Eq. 3.16
• recall coeff. of thermal expansion eT (percent density decrease
per C of T increase), with eT =2.7*10-4C-1 near 22C:
dr= -reTdT, so
dh = e dT
T
h
Eq. 3.17
E.g. 300m upper ocean layer warming 3C
dh = 300  2.710-4C-1  3C = 0.24m
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
Section 3.2 Overview
• Atmos: relationship of density to pressure and temperature
from ideal gas law
• Ocean: density depends on temperature (warmer= less dense,
e.g. sea level rise by warming) & salinity (saltier= more
dense).
• Thermal circulations (Fig. 3.5): warm atmospheric column
has low pressure near the surface and high pressure aloft
relative to pressure at same height in a neighboring cold
region. Reason: see Fig. 3.5
• PGF near surface toward warm region; Coriolis force may
affect circulation but warm region tends to have convergence
& rising. e.g.: Walker, Hadley circulations
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
3.3 Temperature Equation
cw dT = Q
dt
Ocean:
Eq. 3.18
heat capacity  change of T with time* = heating
cw - heat capacity of water
Q – heating J/(kg s)
4200 Joule/(kg K)
heating related to net flux Fnet in Wm-2:
[integrate over layer (density r), e.g. surface layer of depth H
difference in fluxes (surface minus bottom)
dT
net-Fnet
rcw H
= Fsfc
bot
dt
*dT/dt following “parcel”
of water
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
Eq. 3.19 ]
3.3 Temperature Equation
Atmosphere:
dT
dp
1
_
cp
= Q
dt r dt
Eq. 3.20
Similar but new term: -work of expanding
parcel as p decreases
cp - heat capacity of air at constant p.
As in chpt 2:
Q - heating
Q = QSolar+ QIR+ Qconvection+ Qmixing
Eq. 3.21
Heating integrated over the column:
difference in fluxes (top of atm. minus sfc.),
[e.g.:
top
dp
top _
sfc
Q
F
F
∫sfc IR g = IR
IR
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
Eq. 3.22]
3.3.3 Dry Adiabatic lapse rate
Adiabatic:
no heating
0
dT
dp
1
_
cp
= Q
dt r dt
Eq. 3.20
•Term due to work of expansion important to convection:
T decreases as parcel rises (in height) since p decreases
•T decreases even though no exchange of heat with
environment (negligible loss or gain, since parcel rises fast)
•T decrease with height (“lapse rate”) 10 C/km for
adiabatic rising/sinking (no condensation: “dry adiabatic”)
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
Thermals
Example of rising convective
parcel without cloud
• Many parcels moving with dry
adiabatic lapse rate [10C/km]
• Mixing
surrounding air
(“environment”) is brought to
approx same lapse rate*
*Why you can ski in the morning
and surf in the afternoon
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
Figure 3.6
Time derivative following the parcel
dT
• Where are transports? Hidden in
[along with much
dt
complexity]
dT
Total derivative: following an air parcel as it moves
dt
dT
• For local change of T at fixed location, expand
to
dt
show transports by wind
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
Time derivative following the parcel (cont.)
dT
∂T
∂T
∂T
∂T
=
+u
+v
+w
∂t
∂x
∂y
∂z
dt
Eq. 3.27
∂T Local time derivative at fixed place
∂t
∂T
∂T
∂T “Advection” terms: carry properties
u
+v
+w
∂x
∂y
∂z from one region to another
dT
e.g. wind from west u and
= 0 (temp of air parcel
dt
constant), then the local temp change is given by :
∂T
∂T
–u
=
∂t
∂x
…cold air to west gives local temperature dropping
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
Figure 3.7
Initially simple air parcels being deformed in wind fields
• Initially simple patterns become complex [these then feedback
on wind field]
• Yields chaotic motions
• Slight changes in initial conditions yield large changes later
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
Section 3.3 Overview
• Ocean: time rate of change of temperature of water parcel given
by heating
• for a surface layer: net surface heat flux from the atm. minus the flux
out the bottom by mixing
• Atmosphere: Temperature eqn. similar to ocean but…
• when an air parcel rises, temperature decreases as parcel expands
towards lower pressure.
• Quickly rising air parcel (e.g. in thermals): little heat is exchanged
• temperature decreases at 10 C/km (the dry adiabatic lapse rate).
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
Section 3.3 Overview (cont.)
• Time derivatives following parcel hide complexity of the system :
the parcels themselves tend to deform in complex ways if followed
for a long time.
•Results in the loss of predictability for weather.
• The time derivative for temperature at a fixed point is obtained
by expanding the time derivative for the parcel in terms of
velocity times the gradients of temperature (advection).
• Similar procedure applies in other equations.
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
3.4 Continuity equation
Diverging motions
3-D
divergence
Figure 3.7
e.g., ocean sfc.
• Conservation of mass: mass = density • volume
• Divergence in 3 dimensions D3D
rate of change of volume
would tend to reduce density
• Ocean case: hard to change density much
• Horizontal divergence balanced by vertical motions
}
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
3.4 Continuity equation
[Details: full eqn.
dr
= -rD3D
dt
Eq. 3.28]
3.4.1 Oceanic continuity equation
Horizontal divergence:
Ocean approx.
∂u ∂v
D=
+
∂x ∂y
∂w
D=∂z
Eq. 3.30
Eq. 3.29
• Horizontal divergence balanced by net inflow in vertical
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
3.4.2 Atmospheric continuity equation
• Atmospheric continuity eqn.: simple in pressure coord.
• recall pressure surfaces are mass surfaces
• Horizontal divergence D along pressure surfaces must be
balanced by vertical motion
[
_ ∂
D=
∂p
•  vertical velocity in pressure coord.
Eq. 3.31
]
•e.g., low level convergence balanced by rising motion
[Note pressure increases downward so  is negative for rising motions]
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
Coastal upwelling: e.g., Peru; northward
wind component along a north-south coast
• Drag of wind stress tends to accelerate currents northward
• Coriolis force turns current to left in S. Hem
drag
[momentum eqn. fu ≈ Fy ]
• u away from coast
horizontal divergence
from below [thru bottom of surface layer ≈ 50m]
∂w
_
[Continuity eqn. D =
]
∂z
upwelling
Figure 3.9
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
Figure 3.10
Processes leading to equatorial upwelling
• Wind stress accelerates currents westward
[wind speed fast relative to currents, so frictional drag at
surface slows the wind but accelerates the currents]
• Just north of Equator small Coriolis force turns current
slightly to right (south of Equator to the left)  divergence in
surface layer  balanced by upwelling from below
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
Supplementary Fig.: 1998 Annual Ocean Color (Chlorophyll est.)
NASA/Goddard Space Flight Center and ORBIMAGE, SeaWiFS Project
• In tropics, high biological productivity in equatorial cold tongue
and coastal regions due to upwelling of nutrients
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
3.4.5 Conservation of warm water mass
in idealized layer above thermocline
Figure 3.11
warm less dense
cold dense
• Warm light water above thermocline at depth h
• Horizontal divergence/convergence in upper layer
movement of
thermocline
^
∂h
[approx.
+ HD = 0 H = mean thermocline depth,
∂t
^
D = vertical avg. thru layer ]
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
Section 3.4 Overview
• Conservation of mass can be expressed as horizontal divergence
or convergence (of currents or winds) being balanced by changes
in the vertical motion. In the atmosphere this holds in pressure
coordinates.
• Equatorial upwelling results from divergence of ocean surface
currents away from the equator. Water must rise from below to
compensate. The divergence at the equator occurs due to effects
of easterly winds and the Coriolis force (see Figure 3.10).
• For the layer of warm water above the thermocline, convergence
of upper ocean currents* implies a deepening of the thermocline
(see Figure 3.11).
*Note upper ocean layer above thermocline typically deeper than surface layer
in tropics; equatorial upwelling can occur even while thermocline deepens
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
3.5 Conservation of mass applied to moisture
Moisture and Salinity Equations
• conservation of mass: water vapor in atm, salt + water in ocean
dq
= Sources - Sinks
dt
Specific humidity, q =
mass water vapor
total mass air
mass of salt
Salinity, s =
total mass water
• Similarly, for snow models, ice sheets, land hydrology keep track
of water mass per unit area
• Sink from one can be a source for another: e.g., precipitation is a
sink of water substance from the atmosphere, but a source term at
the surface of the ocean & land surface; vice versa for evaporation
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
Links with energy budget:
•Keep track of mass of water vapor, liquid water, snow/ice
separately because of latent heat
• Latent heat of condensation 2.50106 J kg-1;
• Latent heat of of freezing 3.34105 J kg-1
3.5.3 Application: surface melting on an ice sheet
Suppose (during melting season) surface temperature for a region on an
ice sheet remains at freezing so additional increment of downward heat
flux assoc. with global warming, say 5 W m−2, is used for melting.
Rate of decrease of ice thickness =
5 W m−2 (Lf ρi)−1 (365 day/yr × 86400 s/day)
≈ 0.5m/yr. So roughly millennial scale to melt 1.5 km
where density of ice, ρi = 0.9 × 103 kg m−2
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
Section 3.5 Overview
• Conservation of mass gives equations for water vapor
(atmosphere) and salinity (ocean).
• The main sinks of water vapor are due to moist convection
(resulting in precipitation) which at the same time produces
convective heating in the temperature equation (from
condensation in clouds). The main source of water vapor is
evaporation at the surface (it is then transported, mixed, etc). These
processes involve small scale motions and must be parameterized.
• Salinity at the ocean surface is increased by evaporation and
decreased by precipitation.
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
3.6 Moist Processes
Figure 3.12
• At saturation: equilibrium of water vapor with liquid water;
molecules evaporating = molecules condensing
• Unsaturated air tends to conserve water vapor concentration
• Saturation value increases with temperature, so can saturate by
reducing T. Above saturation condensation occurs.
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
Figure 3.12
100%
85%
65%
actual water vapor*
Relative humidity =
saturation value*
[*in units of vapor pressure: converts to specific humidity using total air pressure]
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
3.6.2 Saturation in convection;
lifting condensation level
Recall dry adiabat has no heat exchange but T drops with height
due to expansion
cloud base
Figure 3.13
(lower part)
• Parcel conserves moisture  saturates when T gets cold enough
[lifting condensation level = cloud base]
• As rises further, condensation occurs
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
3.6.3 The moist adiabat and lapse rate in convective regions
Figure 3.13
• As saturated parcel continues to
rise:
• Decrease in p
decrease in T
more water vapor condenses
T drops less than for unsaturated
parcel (roughly 6 C/km as opposed to 10
C/km)
• Process still adiabatic [parcel not
exchanging heat with environment]:
moist adiabatic process
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
Figure 3.13 (cont.)
• If cold parcel rising along “moist
adiabat” is warmer than
surrounding air it continues to rise
[until it reaches level where it is no
longer bouyant]
• Rising parcels warm troposphere
through deep layer [to temperature
close to moist adiabat]: T set by warm
moist surface air
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
Section 3.6 Overview
• Saturation of moist air depends on temperature according to
Figure 3.12. Relative humidity gives the water vapor relative to
the saturation value.
• A rising parcel in moist convection decreases in temperate
according to the dry adiabatic lapse rate until it saturates, then
has a smaller moist adiabatic lapse rate. The temperature curve
in Figure 3.13 (the moist adiabat) depends on only the surface
temperature and humidity where the parcel started.
• If this curve is warmer than the temperature at upper levels,
convection typically occurs.
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
3.7 Wave Processes in the Atmosphere and Ocean: Overview
• Waves play an important role in communicating effects from one
part of the atmosphere to another.
• Rossby waves depend on the beta-effect [change of coriolis force
with latitude]. Their inherent phase speed is westward. In a
westerly mean flow, stationary Rossby waves can occur in which
the eastward motion of the flow balances the westward
propagation. Stationary perturbations, such as convective heating
anomalies during El Nino, tend to excite wavetrains of stationary
Rossby waves.
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
Figure 3.14
Rossby wave westward propagation (mean wind is zero)
[If Coriolis param. f
were const., low
pressure region
could be stationary;
winds circulating in
balance with PGF]
[Increasing f
northward (N. Hem.)
implies imbalance in
mass transport, yields
progagation]
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP
Figure 3.15
Typical Rossby wave pattern excited by a stationary source
Neelin, 2011. Climate Change and Climate Modeling, Cambridge UP