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

Chapter 3Physical Processes in the Climate System

3.1 3.1 Conservation of MomentumConservation of Momentum

3.2 3.2 Equation of StateEquation of State

3.3 3.3 Temperature EquationTemperature Equation

3.4 Continuity 3.4 Continuity EquationEquation

3.5 Moisture and Salinity 3.5 Moisture and Salinity EquationEquation

3.6 Moist Processes3.6 Moist Processes

3.7 Wave Processes in the Atmosphere and Ocean3.7 Wave Processes in the Atmosphere and Ocean

3.8 3.8 OverviewOverview (equals shortcut/level of “need to know”) (equals shortcut/level of “need to know”)

Neelin, 2011. Neelin, 2011. Climate Change and Climate Modeling, Climate Change and Climate Modeling, Cambridge Cambridge UPUP

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. Neelin, 2011. Climate Change and Climate Modeling, Climate Change and Climate Modeling, Cambridge Cambridge UPUP

3.1 3.1 Conservation of MomentumConservation of Momentum

ma = F

•Acceleration a = rate of change of velocity

F - Forcem - Massa - Acceleration

velocity = Coriolis+PGF+gravity+Fdrag eqs. 3.4 & 3.5d dt

•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.

(Eq. 3.1)

Fm=a Use force per unit mass for atm/oc.

Newton:


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 , Velocity components u, v, w (eastward, northward, up).

Figure 3.1


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)

Figure 3.2


Coriolis force animation


• 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]

Coriolis force (cont.)


Coriolis force (cont.)

• is proportional to cos(latitude)

• Always positive and maximum at the equator

• matters because Coriolis so important

“Beta effect” change of Coriolis force with latitude

dfdy


Pressure Gradient Force: e.g., pressure p decreasing eastward

px

[Pressure: force per unit area. Change p across distance x gives force per volume. High to low, so – p/ x .Divide by density (mass per volume) to get force per mass.]

Figure 3.3


Pressure Gradient Force, PGF

• partial derivatives: pressure change eastward and northward, respectively• density (mass per unit volume)

• Force per unit mass, tends to accelerate air from higher to lower pressure

• x direction

• y direction

1

_ px

1

_ py

px


Horizontal momentum equations

velocity = Coriolis+PGF+gravity+Fdrag Eq. 3.2d dt

_ 1

py

dvdt

_= fu + Fdragy

x-direction (east)

y-direction (north) Eq. 3.5

Largest terms(usually)

(Horizontal = perpendicular to gravity)

_ 1

px

dudt = fv + Fdrag

x Eq. 3.4

Note: PGF depends on what’s happening to either side;neighboring regions affect each other


Schematic of geostrophic wind and wind with frictional effects

1fug

_ py=

fvg

1

px=

Geostrophic balance:At large scales at mid-latitudes and approaching the tropics the Coriolis force and the pressure gradient force are the dominant forces

Figure 3.4


Pressure-height relation: Hydrostatic balance

dpdz

_= g

• 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

g dz]

Eq. 3.8

*except in small scale features (thunderstorms, squall lines, etc.)


•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.

Section 3.1 OverviewSection 3.1 Overview


=p

RT

• density 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 T

(percent density decrease per C of T increase), i.e. 0= -0 T(T-T0), with T =2.710-4C-1 for refc temp T0=22C

3.2 3.2 Equation of StateEquation of State

Eq. 3.10

• Relates density to temperature T, pressure p (+other factors)• For the atmosphere: Ideal gas law


UCLA (sea breeze) SM Bay Tropics(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)

Application: thermal circulationFigure 3.5

e.g.:


• 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


-h=h

• recall coeff. of thermal expansion T (percent density decrease

per C of T increase), with T =2.7*10-4C-1 near 22C: = -TT, so

= TT

E.g. 300m upper ocean layer warming 3Ch = 300 2.710-4C-1 3C = 0.24m

Eq. 3.16

Application: sea level rise by thermal expansion

• mass of ocean: area*h. If mass & area constant, while density decreases by , depth h must change by h

Eq. 3.17hh


• 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



Eq. 3.19 ]

heat capacity change of T with time* = heating


Ocean: = QdTdt

cw Eq. 3.18

heating related to net flux Fnet in Wm-2:

[integrate over layer (density e.g. surface layer of depth H difference in fluxes (surface minus bottom)

cw - heat capacity of water Q – heating J/(kg s)

4200 Joule/(kg K)

*dT/dt following “parcel” of water

= FnetFnetdTdt

cw Hsfc bot


cp - heat capacity of air at constant p. Q - heating


As in chpt 2: Q = QSolar+ QIR+ Qconvection+ Qmixing

Eq. 3.20

Eq. 3.21

∫top

sfc

QIRdpg = FIR

top FIRsfc_ Eq. 3.22]

Heating integrated over the column: difference in fluxes (top of atm. minus sfc.), [e.g.:

Atmosphere:

work of expanding parcel as p decreases

Similar but new term:

= Q1

dTdt

cpdpdt

_


dpdt

_

3.3.3 3.3.3 Dry Adiabatic lapse rateDry Adiabatic lapse rate

= Q1

dTdt

cp Eq. 3.20Adiabatic: no heating

•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”)

0


Thermals

• Many parcels moving with dry adiabatic lapse rate [10C/km]

• Mixing surrounding air (“environment”) is brought to approx same lapse rate*

Example of rising convective parcel without cloud

*Why you can ski in the morning and surf in the afternoon

Figure 3.6


Time derivative following the parcel

dTdt

Total derivative: following an air parcel as it moves

• Where are transports? Hidden in [along with much complexity]

dTdt

• For local change of T at fixed location, expand to show transports by wind

dTdt


Time derivative following the parcel (cont.)

“Advection” terms: carry properties from one region to another

e.g. wind from west u and = 0 (temp of air parcel

constant), then the local temp change is given by :

∂T∂t = – u

∂T∂x

Eq. 3.27

∂T∂t

Local time derivative at fixed place

dTdt

=∂T∂t

+ u∂T∂x

+ v∂T∂y

+ w∂T∂z

u∂T∂x

+ v∂T∂y

+ w∂T∂z

dTdt

…cold air to west gives local temperature dropping


• Initially simple patterns become complex [these then feedback on wind field]

• Yields chaotic motions• Slight changes in initial conditions yield large changes later

Initially simple air parcels being deformed in wind fieldsFigure 3.7


• 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).



• 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.

Section 3.3 Overview (cont.)Section 3.3 Overview (cont.)


Diverging motions

3.4 3.4 Continuity equationContinuity equation

• 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

3-Ddivergence e.g., ocean sfc.

}

Figure 3.7


∂v∂y

∂u∂x

+D =

3.4 3.4 Continuity equationContinuity equation

ddt

= -D3D[Details: full eqn. ]

Horizontal divergence: Eq. 3.30

∂w∂z

D = -Ocean approx. Eq. 3.29

Eq. 3.28

• Horizontal divergence balanced by net inflow in vertical

3.4.1 3.4.1 Oceanic continuity equationOceanic continuity equation


3.4.2 Atmospheric 3.4.2 Atmospheric continuity equationcontinuity 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

Eq. 3.31

• vertical velocity in pressure coord. ]

•e.g., low level convergence balanced by rising motion

[Note pressure increases downward so is negative for rising motions]

∂∂pD = _[


• Drag of wind stress tends to accelerate currents northward

• Coriolis force turns current to left in S. Hem

[momentum eqn. ] • u away from coast horizontal divergence upwelling from below [thru bottom of surface layer ≈ 50m]

[Continuity eqn. ]

Coastal upwelling: e.g., Peru; northward wind component along a north-south coast

Figure 3.9

fu ≈ Fydrag

∂w∂zD = _


Processes leading to equatorial upwelling

• 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

• Wind stress accelerates currents westward [wind speed fast relative to currents, so frictional drag at surface slows the wind but accelerates the currents]

Figure 3.10


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


Figure 3.11

∂h∂t

+ HD = 0^

3.4.5 3.4.5 Conservation of warm water massConservation of warm water massin idealized layer above thermoclinein idealized layer above thermocline

cold densecold dense

warm less densewarm less dense

• Warm light water above thermocline at depth h

• Horizontal divergence/convergence in upper layer movement of thermocline

[approx. H = mean thermocline depth, D = vertical avg. thru layer ]^


• 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



3.5 Conservation of mass applied to moisture3.5 Conservation of mass applied to moistureMoisture and Salinity Moisture and Salinity EquationsEquations

Specific humidity, q =mass water vapor

total mass air

Salinity, s =mass of salt

total mass water

• conservation of mass: water vapor in atm, salt + water in ocean

• 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

= Sources Sinksdqdt


3.5.3 Application: surface melting on an ice sheet3.5.3 Application: surface melting on an ice sheet

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

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.

Links with energy budget: Links with energy budget:

•Keep track of mass of water vapor, liquid water, snow/ice separately because of latent heat

• Latent heat of condensation 2.50106 J kg-1;

• Latent heat of of freezing 3.34105 J kg-1


• 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.



Figure 3.12

3.6 Moist Processes3.6 Moist Processes

• 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.


Relative humidity =

saturation value*

actual water vapor*

[*in units of vapor pressure: converts to specific humidity using total air pressure]

65%65%

85%85%

100%100%

Figure 3.12


Figure 3.13 (lower part)

3.6.2 Saturation in convection; 3.6.2 Saturation in convection;

lifting condensation levellifting condensation level

Recall dry adiabat has no heat exchange but T drops with height due to expansion

• Parcel conserves moisture saturates when T gets cold enough [lifting condensation level = cloud base]

• As rises further, condensation occurs

cloud base


Figure 3.13

3.6.3 The moist adiabat and lapse rate in convective regions3.6.3 The moist adiabat and lapse rate in convective regions

• 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


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


• 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.



• 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.

3.7 Wave Processes in the Atmosphere and Ocean: Overview3.7 Wave Processes in the Atmosphere and Ocean: Overview


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]


Figure 3.14

Typical Rossby wave pattern excited by a stationary source

Figure 3.15


Documents

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