Atmospheric Science 8600 Advanced Dynamic Climatology
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Atmospheric Science 8600 Advanced Dynamic Climatology

Anthony R. Lupo. 302 E ABNR Building. Department of Soil, Environmental, and Atmospheric Science . University of Missouri . Columbia, MO 65211. Phone: 573-884-1638 . Email: . LupoA@missouri.edu. Web.missouri.edu/~.

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Atmospheric Science 8600 Advanced Dynamic Climatology




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Presentation on theme: "Atmospheric Science 8600 Advanced Dynamic Climatology"— Presentation transcript:

Slide1

Atmospheric Science 8600 Advanced Dynamic Climatology

Anthony R. Lupo

302 E ABNR Building

Department of Soil, Environmental, and Atmospheric Science

University of Missouri

Columbia, MO 65211

Phone: 573-884-1638

Email:

LupoA@missouri.edu

Web.missouri.edu/~

lupoa

/atms8600.html or atms8600.pptx

Slide2

ATMS 8600 Dynamic Climatology

Syllabus **

1. Introductory and Background Material for ATMS 8600.

2. Define Climate and climate change.

3. The Equations of climate processes

4. Physical processes involved in the maintenance of climate.

Slide3

ATMS 8600 Dynamic Climatology

5.

Climate

Modelling

, a history and the present tools.

6. Climate Variations: Past climates and what controls climate?

7. Climate change: Inductive and deductive theories.

8. Climate and Climate change (natural vs. “anthropogenic”, what is the current conventional wisdom?)

** Students with special needs are encouraged to schedule an appointment with me as soon as possible!

Slide4

ATMS 8600 Dynamic Climatology

This class will essentially just go 'upscale' from ATMS 8400 - Theory of the General Circulation.

Some initial questions:

What is Climate? How does it differ from weather (synoptic, or even

the General Circulation)?

How is this different from Climatology?

Weather

the day to day state of the atmosphere. Includes state

variables (T, p)

and descriptive material such as cloud cover and

precipitation amount and type,

etc.

Slide5

ATMS 8600 Dynamic Climatology

Recall in meteorology, we tend to divide phenomena by scale based on what processes are important to driving them.Table 1

Scale

Time

Space

Force

Weather

Planetary

7

– 14 days

6000+ km

Co, PGF

Jet stream

Synoptic

1 – 7 days

2000 – 6000 km

Co, PGF,

Fric

,

Bouy

Low pressure

Meso

1

- 24 h

10 – 2000 km

Bouy

, PGF,

Fric

Fronts

Micro

< 1

hr

< 10 km

Bouy

,

Fric

, PGF

Thunderstorm

Slide6

ATMS 8600 Dynamic Climatology

General Circulation – ‘statistical’ features. We think of planetary in scale, but time scales are 2 weeks, 1 month, 1 season, 1 year, a few years.

Climate

Is the

long-term

or

time mean state

of the

Earth-

Atms

. system

and the state variables along with higher order statistics. Also, we must describe extremes and recurrence frequencies.

Thus, the general definition of climate is scale independent and a technical definition would depend on your scale,

ie

we can describe micro,

meso

, and synoptic climates and

climatologies

.

Climate

as we will discuss it many contexts will be "global" or large-scale in nature, or "upscale"

in time from

the

General Circulation.

Slide7

ATMS 8600 Dynamic Climatology

Climatology is the study of climate in a mainly descriptive and a statistical sense. Climatologists study these issues.

Dynamic Climatology

or

Climate dynamics

are

relatively

new concepts and involve the study of climate in a theoretical and/or numerical sense. In order to study climate in this sense, we will use models, which will be derived using basic equations.

One

way to accomplish this is via the scaling of primitive equations, or using basic 'RT' equations (Energy Models, which use concepts like

Stefan

Boltzman's

law).

Slide8

ATMS 8600 Dynamic Climatology

Key concepts that will be discussed in this course:

We'll

need to

distinguish

plainly between weather and climate! (Review the concept of climatic averaging')

We'll

need to talk about time variations on climate states ('climate' versus climate change)

We'll

need to examine the components of the climate system

Slide9

ATMS 8600 Dynamic Climatology

We'll

need to examine the state of the climate, in particular this means 'internal variables' (internal vs. external).

We'll

need to study climatic 'forcing' (this means 'external variables (e.g.

Solar, Plate tectonics, humans?), (also how

they differ from internal))

We'll

need to examine issues surrounding spatial resolution and climatic character.

Slide10

ATMS 8600 Dynamic Climatology

The primary components of the Climate system

1. The Atmosphere (typical response time --> minutes to

three weeks)

2. The Ocean (typical response time months and years, for upper ocean

)

3. The

litho

-biosphere (we'll treat as one for now

)

4. The

cryosphere

(both land and sea ice, response times on order of decades to MYs)

Slide11

ATMS 8600 Dynamic Climatology

The earth-atmosphere system, courtesy of Dr. Richard Rood.(http://aoss.engin.umich.edu/class/aoss605/lectures/)

Slide12

ATMS 8600 Dynamic Climatology

Aside:

Typical short-hand notations used now in the study of climate:

10,000 Years Before present (10 KY BP) KY = Thousands of years BP = before present

MY = Millions of years.

Slide13

ATMS 8600 Dynamic Climatology

Another view of the climate system

Slide14

ATMS 8600 Dynamic Climatology

Another view

Slide15

ATMS 8600 Dynamic Climatology

Each component

of the climate system can

be described by it's own state variables, which are considered internal variables

.

External

forcing is defined as forcing outside the system or sub-system. Thus, SST anomalies are internal or state variables for the earth atmosphere system, or the ocean. But they are considered 'forcing' or external to the atmospheric component

.

Also, the dynamics of the internals are fairly well know, but heat and mass exchange processes between sub-

sytems

not well understood.

Slide16

ATMS 8600 Dynamic Climatology

Ok, now we have to introduce ourselves to the concept of climatic averaging. For any instantaneous climatic variable a; a (bar) representing a time mean, and a' the instantaneous departure from the mean.

Slide17

ATMS 8600 Dynamic Climatology

Recall, that (from mean value theorem):where t' will represent a 'dummy' time coordinate, and t is physical time, and t is the averaging time usually written as =t . For the above to be valid in a physical sense, then: P(a') <<  << P(a)

Slide18

ATMS 8600 Dynamic Climatology

where P(a) represents a characteristic time period, say a few seconds for wind gusts (P(a')?), and 3 - 4 days for an extratropical cyclone (P(a)?)

Slide19

ATMS 8600 Dynamic Climatology

In drawing on the previous slide, I've provided for you a CLEAR separation between the two periodic fluctuations.In the climate system one must look for periods of high and low variability, to do this we can look at an idealized (not real) periodogram for the atmosphere. Periodogram (real) examines the "power" spectrum within a time series of say, Temperature. "power" or variance is the square of the Fourier coefficients:

Slide20

ATMS 8600 Dynamic Climatology

An idealized Temperature Spectrum for Earth

Slide21

ATMS 8600 Dynamic Climatology

This spectrum demonstrates 'scale' separations nicely (planetary, synoptic,

meso

, micro):

typical

averaging

periods:

example: 1

sec, 1

hr

, 3-12 mo., 30 years, 1 MY

Turbulence

way

out on right

'weather' and gen circ.

12

hr

to 1

yr

typically.

climate (as is commonly thought of) is

1

yr

to 27

year

peak.

climate change scale:

30

yrs

to 100 KYs.

tectonic change: on left.

Slide22

ATMS 8600 Dynamic Climatology

Forcing:

External forcing

Is “boundary value” forcing. These are independent of the system and can be altered externally. These refer to forcing outside the system or outside the sub-system is it is closed. Example: solar radiation.

Internal forcing

are

forcings

that operate within the system and can arise out of non-linear interactions within a system. Example:

Vorticity

advection, temperature advection, latent heating.

Slide23

ATMS 8600 Dynamic Climatology

The

State

of the Climate

System

In

the climate system can be considered a composite system, if as a whole the system is thermodynamically closed (impermeable to mass, but not energy), (recall from gen circ., we say mass does not change

).

The

individual sub-components are thermodynamically open (transfers of mass and energy allowed) and cascading (that is the output of mass or energy from one subsystem becomes the input into another).

The state of the climate system can be represented in terms of physical variables that represented

additive

of extensive properties (e.g., volume or internal energy, or

angular

momentum), or in terms of intensive properties ('fields') that are independent of total mass and that change with time (

'temperature',

pressure, and wind velocity).

Slide24

ATMS 8600 Dynamic Climatology

Equilibration of the Climate System

If

for specific time scales, the internal climate system behaves as if it has forgotten its past, and responds primarily to external forcing then it can be considered to be almost in the state of equilibration.

External

forcing can be outside earth atmosphere. system, or be forcing from an internal variable of longer time-response (inertial time scale) sub-systems forcing on another sub-system (e.g. SST forcing) with a shorter time response. (e.g., if considering synoptic-scale, then ice sheets, oceans, and everything is 'external').

Slide25

ATMS 8600 Dynamic Climatology

Thus the climate system is a boundary value problem, this is different from weather forecasting with is primarily an initial value problem (boundaries there too). These definitions allow us to define climate in terms of ensemble means and variability of each sub-system independently

.

If

any initial state always leads to the same near equilibrium climatic state (same

equilibrium

properties), then the system is transitive (climate folks) or

ergodic

(

geology

folks!).

Slide26

ATMS 8600 Dynamic Climatology

Transitive (weather forecasts, “cycles” diurnal and annual)

Slide27

ATMS 8600 Dynamic Climatology

If instead there are two or more

different

states with

different properties that

result from different initial conditions, then the system is intransitive (this

system – stochastic, dynamic laws unknown - pack

it up and go home w/r/t forecasting).

If

there are different subsets of statistical properties, which a transitive system assumes during its evolution from different initial states, through long but finite periods of time, the system is almost intransitive.

In

this case, the climate state, beginning from any initial condition will always converge to the same state eventually, but go through periods w/ distinctly different climatic regimes. This is best representation of climate system. (Ice ages?)

Slide28

ATMS 8600 Dynamic Climatology

Intransitivity (weather forecasts) and “almost intransitive” (oscillations);

Slide29

ATMS 8600 Dynamic Climatology

Concept: Diagnostic vs. PrognosticQuick definition: prognostic equations have a on the Left hand sideDiagnostic equations have no time derivatives.Example: Equation of state (ideal gas law).

Slide30

ATMS 8600 Dynamic Climatology

A more precise (correct) definition: source sinkThis is a standard ordinary differential equation, or Forced – Dissipative equation/system. This is nothing new.

Slide31

ATMS 8600 Dynamic Climatology

F = any external forcing (source) = damping constant (sink) This is one way to view long-term climate change. If  < =  (damping is large, strong or instantaneous) we have a diagnostic, or equilibrium problem such that (the two RHS terms cancel or nearly so;

Slide32

ATMS 8600 Dynamic Climatology

If  >>  and  >= P(a) then we have a prognostic equation or non-equilibrium problem, e.g., These properties allow us to distinguish based on approximate values of (), fast response variables is diagnostic and slow response variables are prognostic variables in climate system.

Slide33

ATMS 8600 Dynamic Climatology

For climate:

fast response (diagnostic)

atmosphere

slow response (prognostic)

ice

sheets, ocean

(for example, if

ice sheets

grow, we know how atmosphere will behave)

Slide34

ATMS 8600 Dynamic Climatology

Diagnostic or prognostic?

Slide35

ATMS 8600 Dynamic Climatology

Climate Modeling

Definition of a Climate Model:

An hypothesis (

frequently

in the form of mathematical statements) that describes some process or processes we think are physically important for the climate and/or climatic change, with the physical consistency of the model formulation and the agreement w/ observations serving to 'test' the hypothesis (i.e., the model).

"The model (math) should be shortened (approximated) for testing the hypothesis, and the model should jive with reality".

Slide36

ATMS 8600 Dynamic Climatology

The scientific Method:

1. Collect Data

2. Investigate the Issue

3. Identify the Problem

4. Form Hypothesis

5. Test Hypothesis

6. Accept or Reject hypothesis based on conclusions

7. If reject,

goto

2

8. If accept, move on to the next problem.

Slide37

ATMS 8600 Dynamic Climatology

Two distinct types of climate models:

1) Diagnostic or equilibrium model (

Equilibrium Climate Model - ECM)

with time derivatives either implicitly or explicitly set to zero. The ECM is most commonly solved for climatic means and variances

. (

d /

dt

= Force + Dissipation)

2) Prognostic models, where time derivatives are crucial and with the variation with time of particular variables the desired result (i.e., a time series). Most commonly solved for changes in climatic means and variances

. Weather models (General Circulation Models – GCMs)

Slide38

ATMS 8600 Dynamic Climatology

Ocean - Atmosphere (one of many subsystems of the climate system)

(go back

to our diagram)

Some basic principles:

Conservation of mass

precipitation, evaporation:

precip

=

evap

over the globe (closed system). Water vapor budget equation

we also use in

Atms

8400.

Energy: Sun

heats land and oceans which, in turn, heats the atmosphere (transparent to

shortwave

, but opaque to LW).

In order to fully understand, we should couple the

atmosphere - ocean

is more important than considering each separately, however, we know each separately!

Slide39

ATMS 8600 Dynamic Climatology

Feedback Mechanisms

Feedback mechanisms complicate things, Nature is highly non-linear. But they are a good way to get a handle on non-linear coupling mechanisms.

Feedback mechanisms are all

positive (+)

or

negative (-)

(+) amplify the linear response to a forcing process. (e.g., ice-albedo

) – indicate a “sensitive” system.

(-) de-amplify the response to a forcing process. (e.g

., clouds – global temperature)

Slide40

ATMS 8600 Dynamic Climatology

Examples: Double

CO

2

-Positive feedback-

Linearly increase

CO

2

to double the amount has a very small effect on climate temp. response or forcing

.

It's

the other feedbacks that amplify "global warming"

CO

2

is a "potent" greenhouse gas, but a trace gas.

Slide41

ATMS 8600 Dynamic Climatology

Water

Vapor is more plentiful and 30 times more potent "greenhouse gas" (molecule per molecule, due to vibrational, rotational absorption bands).

Increased CO

2

,

(could) mean increased water vapor. This increases the long wave retained by

atmosphere,

which heats up

atmosphere

and oceans

. (IPCC)

Increased

water vapor

increased

LW heating,

increased

evaporation

increased

water vapor. (Get the picture? Positive feedback!). This will go on forever, unless something interrupts.

Caveat

:

Increases in

air temp, does not necessarily mean corresponding

increases

in

specific humidity

and

dew points

.

Formula for + feedback:

E happens

A

inc

B

inc

C

inc

A

inc.

Slide42

ATMS 8600 Dynamic Climatology

-- negative feedback --

Clouds

again same caveat, more vapor not necessarily means more clouds, and consider cloud characteristics, like droplets, or ice crystals, drop size, optical depth, etc.

More low clouds

,

increases

Earth's albedo, less SW into the system,

LW out

exceeds SW in and cooling.

May

dampen a positive feedback, bring the system back into equilibrium, but not necessarily at the same state that was the original.

Formula - feedback:

E happens

A

inc.

B

inc.

C

inc

A

dec.

Slide43

ATMS 8600 Dynamic Climatology

Temperature measurements and records (problems w/ climate data)

Must make objective (homogeneous)!

Climate

statistics

there are large

imhomogeneties

in time and space for recorded measurements (station moved?,

instrumentation

changed? obs. practices changed? land surface change?). These are hard to get a handle on in reality? Does observation match reality?

Many

problems

exist

!

"to make these records homogeneous, we have to choose an objective weighting scheme". However, choice is highly objective!! Skeptics can see no change, proponents can see change

Slide44

ATMS 8600 Dynamic Climatology

How do we generate such records? Past Variations of Climate!

1. What do we really mean?

2. How do we reconstruct past climates?

3. How do we infer past variations in climate?

4. Types of climatic data?

In order to "see" past climate you must understand present!

Slide45

ATMS 8600 Dynamic Climatology

3 types of climate data

Observed

observed

data, there is about 350 years for England, 200 years in the West, 15 to 50 years globally.

Historical

based on historical recordings, diaries, paintings, etc. Most is qualitative and uncertain, but we have this back 100's to 1000's of years.

Proxy

infer climate, via chemical biological, and sediment records.

Slide46

ATMS 8600 Dynamic Climatology

Typically proxy records involve examining pollen/spore records in sediment or fossils, and/or matching plant and animal species with

current

climate types.

We

can also infer climate from isotope records, for example examining Carbon-13 or Oxygen-18 isotopes. Plant different plant species use C-13 differentially, thus it is easy to tell what species persisted in some area by examining remains.

O-18. There is more O-18 in the oceans under colder climes (the molecules less likely to

evaporation

than lighter ones).

Read the two articles about proxy determination. The approach is largely statistical,

i.e.

we correlate concentrations to

Tavg's

and come up with a regressive relationship.

Slide47

ATMS 8600 Dynamic Climatology

Also

read (Pollen):

Woodhouse, C.A., and J.T.

Overpeck

, 1998: 2000 years of drought variability in the Central United States.

Bull. Amer. Met. Soc

.,

79

, 2693 - 2714.

Slide48

ATMS 8600 Dynamic Climatology

The Fundamental Equations:

Now my favorite part - the equations. Not only will we examine these in a climate sense, but we will examine for a general substance, (fluid) as well. Think of this as 'fluid dynamics'

What we are doing is representing (or evaluating) meteorological, oceanographic, or other relevant observations. Since these observations are taken at discrete intervals of time (say about 6

hrs

for

weather observations),

we assume synoptically

averages

conditions. That

is:

x = X + X* (synoptic mean and dep.)

The equations we'll derive will be general then climatologically

avg'd

.

Slide49

ATMS 8600 Dynamic Climatology

Recall our general conservation laws:

1) Cons. of Mass (vol.) Continuity (Water mass also!)

2) Cons. of momentum (N-S equations)

3) Cons. of Energy (1st law)

and don't forget elemental kinetic theory

of gasses (State

(constituent) variable relationships)

Continuity:

Notation c = carrier fluid a =

atms

. w = ocean i = ice

j = trace

constituents

Slide50

ATMS 8600 Dynamic Climatology

Xc = mass concentration of carrier fluidXj = mass conc. of jth trace constituentTotal density of an arbitrary volume in the climate system Often times, Xc >> Xj

Slide51

ATMS 8600 Dynamic Climatology

Also, to be “pure”, we need to say for example  = Xc / Volumemass fraction or mixing ratio (trace constituents): Remember in Atmospheric Science? Mixing ratio: Mv / Md?

Slide52

ATMS 8600 Dynamic Climatology

Continuity equation (general) carrier see where atmosphere’s. version comes from? Water vapor balance eqn? Continuity?Atmospheric version:Sc  0 on the time scale of days to 100 KY, but on the time-scale of MY Sc doesn’t go to 0!

Slide53

ATMS 8600 Dynamic Climatology

Xc   (atmospheric density - Continuity) (From Atmospheric dynamics)Water Vapor (trace gas):Sc  Source + sinkXc  q (specific humidity Mv / (Md + Mv)

Slide54

ATMS 8600 Dynamic Climatology

trace continuity equation Vc = velocity of carrierVj = velocity of traceSc = sources sinks of carrierSj = sources sinks of trace

Slide55

ATMS 8600 Dynamic Climatology

Recall from vector calculus we can put into advective formsAs in atmospheric version

Slide56

ATMS 8600 Dynamic Climatology

Also, we can decompose Vj as follows where,mj = molecular diffusionwjk - fall due to potential gravity fieldthen,

Slide57

ATMS 8600 Dynamic Climatology

Where, A B CA = turbulent eddy fluxB = mean molecular diffusion fluxC = mean sedimentation

Slide58

ATMS 8600 Dynamic Climatology

Again, water vapor: j = H2Ov Xj =  S = Evap + PrecipOne can get the fundamental equation for water vapor budget (hydrology - hydrologic cycle), which with different choices of symbols, and consider in a column, will give general circulation equation.

Slide59

ATMS 8600 Dynamic Climatology

Finally one could consider the total fluid volume (carrier + trace substances combined!).

Slide60

ATMS 8600 Dynamic Climatology

The Equation of Motion (Conservation of momentum) Essentially a re-statement of the equation of motion from large-scale atms. dynamics. PGF (B) Grav (app) (B) CO (Apparent)where:G = tidal forces (Body)E = external forces (body)F = Friction (stress, surface)V,, p assume their normal meaning

Slide61

ATMS 8600 Dynamic Climatology

In component form (spherical coords)

Slide62

ATMS 8600 Dynamic Climatology

Thermodynamic Equation

It

is of extreme importance for climatic processes that terrestrial state variables (T and P) are close to the triple point of water. This has important implications for earth's climate.

Thermodynamic equation (first Law)

dh = du +

dw

dq

= du +

dw

Slide63

ATMS 8600 Dynamic Climatology

dh (or dq) = diabatic heating, rate of addition of heatdu = rate of change of internal energy per unit massdw = rate of work on unit mass by compression (pressure work term)In the atmosphere: NOT SAME AS

Slide64

ATMS 8600 Dynamic Climatology

In these equations: = 1 /  OK, now we can rewrite the first law:

Slide65

ATMS 8600 Dynamic Climatology

Take total differential of r in flux form  use vector identity flux = adv. + divergence/convergence, and substitute on the RHS (leave flux on RHS and time diff on LHS) and rewrite using general continuity equation (similar to energy equation in Atms 8400!): Then in flux form:

Slide66

ATMS 8600 Dynamic Climatology

We can further conceptually evaluate the diabatic heating (source/ sink) – in climate these processes are of critical importance! Hrad  radiational heatingH cond  conductional heatingH conv  convective heating (Hcond + Hconv) = H sensH lat  phase transformations of any consituent!C  internal energy source (chemical reactions)d  dissipative processes (Climate model, must be general, these models could be ported to any planet).

Slide67

ATMS 8600 Dynamic Climatology

Then, you must relate internal energy du to T This varies for different carriers and trace substances in the climatic system.Note, here we define Liquid water at 273 to be the level of zero enthalpy, this is done in some models. Component form of duDry air H2Ov

Slide68

ATMS 8600 Dynamic Climatology

Component form of duDry air + moist. Liquid water: Ice Lithosphere

Slide69

ATMS 8600 Dynamic Climatology

Equations of state (

Constituentive

Relationships)

Equations of state (elemental kinetic theory) relates the key state variables to each other, for example:

Gasses;

Charles Law (V,T) (Volume, density to temperature) (~1787 AD)

Boyles Law (Pressure, to temperature) (~1600 AD)

Combined Gas law or Ideal gas law (pressure, density, temperature =

Const

) (~1850 AD)

Slide70

ATMS 8600 Dynamic Climatology

This works for any gas, or mixture of gasses, whether on earth or not.

For Atmosphere;

P

/ T = R*/mg = R (where R is a combined gas constant, Recall Dalton's Law again!)

For liquids

Similar relationships are found.

Example: Earth’s Oceans

T,

, P, and Salinity (S)

Slide71

ATMS 8600 Dynamic Climatology

 These relationships are more complicated, higher order non-linearities (dependencies) of density on P,T, and Salinity.There is no theoretical exact expression akin to Ideal Gas Law! However, we can derive polynomial expressions empirically!Oceans: density written as: Oceanography prefers to work with isosteric processes (sea water):

Slide72

ATMS 8600 Dynamic Climatology

(again, look up on tables from polynomial relations):Terms on RHS are empirically derived relationships (weighting factors) as functions of the following;term 1) is isostere at Standard Temp and Pressure and Salitinity (level of Zero enthalpy -- in atmosphere at standard sea level pressure and freezing right?)term 2) is the temperature anomaly as a function of T only

Slide73

ATMS 8600 Dynamic Climatology

term 3) is a salinity anomaly

as a

function of S only

term 4) is a salinity/temp anomaly

as a

function of S and T combined (non linear term)

term 5) is a salinity/pressure anomaly (function of S and P combined)

term 6) is a temp/press anomaly (function of T and P)

term 7) is a S/T/P anomaly (function of all three)

again, highly non-linear relationship!!!

Slide74

ATMS 8600 Dynamic Climatology

Equation of State for ice;

a) for sea ice

we will

have a similar relationship to the ocean.

b) land ice --. resort to glaciological theory. We'll punt on this!!

Slide75

ATMS 8600 Dynamic Climatology

So Review fundamental equations of Geophysical Fluid dynamics:A. Continuity: B. Equation of Motion

Slide76

ATMS 8600 Dynamic Climatology

C. Thermodynamics: D. Equation of State (Gasses)

Slide77

ATMS 8600 Dynamic Climatology

Recall concept of Reynolds Averaging Where X’ is much smaller than XbarInstantaneous wind: (u) (where prime is a departure from time average, say yearly, seasonally, or any time scale appropriate to the phenomena you’re studying.) This averaging can be carried out no matter what scales you examine!

Slide78

ATMS 8600 Dynamic Climatology

u = [U] + u* (where * is a departure from the space average at a given instant, say the average around 40 N.Then by definition; [u*] = 0 Also, Does ? Mathematically, they are interchangeable, but physically it makes more sense to time average first and then space average. This is also convention!

Slide79

ATMS 8600 Dynamic Climatology

So, the product of U and V: So, if we average the product, then? What must happen to middle terms?

Slide80

ATMS 8600 Dynamic Climatology

We can, then naturally apply the same procedure to space averaged data to get:[uv] = [u] [v] + [u* v*] (2)how about?

Slide81

ATMS 8600 Dynamic Climatology

OK, we know what the mean product of u and v is (1), but if we take ubar and vbar and space average their product we get (2) And then space average (1) and substitute (2) for the first term on the right hand side to get (3). (OK, you try it!)

Slide82

ATMS 8600 Dynamic Climatology

What does (3) mean? A B CLHS = total transport of relative linear ZONAL momentum (v is transport wind)Term A = transport by mean meridional circulations (Hadley, Ferrel, and Polar direct cells)Term B = transport by stationary waves or STANDING eddies (Alleutian Low, Bermuda High, monsoon circulations.Term C = transport by transient eddies (travelling distrubances) (e.g. sit at a point and watch the sign change)Global Climatic Balances of Mass, angular Momentum, and Energy (Apply to equations for climate processes). We will not necessarily derive Momentum and Energy relationship since we already did that for Atmosphere in ATMS 8400. We'll just mention them, and use notation common for fluid dynamics where applicable.

Slide83

ATMS 8600 Dynamic Climatology

We also sort of examined mass balance through water vapor, but might be useful to look at again: If we start with mass balance for atmos (consideredno change for Gen Circ.): (dry atms. + water)

Slide84

ATMS 8600 Dynamic Climatology

consider an arbitrary 'box' of climate

system (previous)

Zs

= surface /

atms

. interface which varies over land, but is sea - level (

zsl

) over the ocean

.

Zt

= top and

Zo

= “Lorenz Condition”.

Let's choose an appropriate Boundary Cond. for vertical velocity (w)

w top = 0

Slide85

ATMS 8600 Dynamic Climatology

w surf = 0 for most synoptic and gen. circ. applications. For climate choose (climate is a Boundary Condition problem as opposed to weather forecasting which is an initial value problem): Integrate over atmosphere contained in the box, first over arbitrary column (this is the dry Atmosphere only;

Slide86

ATMS 8600 Dynamic Climatology

assume atmosphere is well behaved, reverse order of integration and differentiation (and assume wt = 0): A B Cwe call dry mass integral (also a,b,c are boundary recall)

Slide87

ATMS 8600 Dynamic Climatology

Where A = Area integral (in latitude (f) and longitude (l))for total mass of box (atmosphere - dry air) Then assume hydrostatic balance;In particular let's do a zonal averaging procedure (e.g. an average around a latitude circle in spherical space

Slide88

ATMS 8600 Dynamic Climatology

First decompose: Then zonally average:Then time average (bar)

Slide89

ATMS 8600 Dynamic Climatology

let’s do it (imagine a “bar” over the [ ] quantity);thus, v and w are mass fluxes if (we can divide [ ]bar into mean motions + eddies – assume conservation ok for a couple centuries!);

Slide90

ATMS 8600 Dynamic Climatology

Global Mass balance of water vapor

Vapor

is of course the most important trace substance

We derive an equation in general circulation, however, we can do this a different way. Also, in climate we’ll look at each sub component of the system and their

interactions, since water mass is everywhere.

Slide91

ATMS 8600 Dynamic Climatology

Start with continuity: we’ll say Xj = Xv (atms. vapor) =Xn (cloud water) =Xw (surface water – land and oceans) =Xi (surface ice)

Slide92

ATMS 8600 Dynamic Climatology

Then for vapor (1), Evap Subli. lv svfor cloud water (2) Cond.  Ablat.

Slide93

ATMS 8600 Dynamic Climatology

Surface water (3) surface ice (4): So, we come up with 4 equations for water in the climate system, one for atms., clouds, surface water (oceans and land – Soils guys), and ice (glacioligist). This is ATMS 8400 General Circulation times 4!

Slide94

ATMS 8600 Dynamic Climatology

As before we can integrate each in the column: use the following notation:example condensation!

Slide95

ATMS 8600 Dynamic Climatology

Then for water vapor equation: horiz and vert cloud surf vapor flx formationRedefine Xv (mass of water vapor mv):

Slide96

ATMS 8600 Dynamic Climatology

Then Redefine horizontal fluxes of water vapor (shorten notation!):and the horizontal surface flux is the topography term, accounting for horizontal motions and vertical flux from surface. These are precip. + evap.:C = Cond. and freezing from clouds (combined E and A from above)

Slide97

ATMS 8600 Dynamic Climatology

Diagram

Slide98

ATMS 8600 Dynamic Climatology

Then (Atmospheric water vapor): precipitable horizontal Evap precip Cloud water transport waterUsing the same procedures as before, we change variables and handwave.

Slide99

ATMS 8600 Dynamic Climatology

Cloud water: transport of precip phase clouds changesSurface Water (Soil science folks like this): Evap Melting liquid precip

Slide100

ATMS 8600 Dynamic Climatology

Distinguish between land and oceanIn the long term, the divergence of J terms will be equal for land and oceans, and they will = Runoff + Underground water. Again, the soil science community like these!

Slide101

ATMS 8600 Dynamic Climatology

Surface Ice (fundametnal cont. equation of glaciology) snow snow snow evap melt Glacier can be considered a liquid and all the dynamics and movement can be accounted for.Shows how ice sheets grow and considers it's movement and "plasticity" (stretching) or deformation!

Slide102

ATMS 8600 Dynamic Climatology

Globally integrate the four and all divergences vanish (transports disappear of course!).

Slide103

ATMS 8600 Dynamic Climatology

Then add 'em all up to show balance:Conservation for water in climate sytem! In Practice: mw >> mi >> mv + mn and mv >> mn mw = 1.400 x 1021 kg mi = 2.0 x 1019 kg and mv + mn = 1.6 x 1016 kg

Slide104

ATMS 8600 Dynamic Climatology

For the Atmosphere we can show the General Circulation version of the water balance equation versus climate (what’s typically used). Dynamic Climates General Circulation

Slide105

ATMS 8600 Dynamic Climatology

We covered this in General Circulation recall primarily done in tropics by mean motions and in the mid-latitudes by eddy motions.

Slide106

ATMS 8600 Dynamic Climatology

Angular momentum BudgetsRecall we considered the earth and atmosphere as separate and then together. This is reasonable to consider angular momentum budgets as a conservational process for scales of decades, but is not appropriate for timescales of thousands of years or longer.The fundamental constraint is:where angular momentum was:

Slide107

ATMS 8600 Dynamic Climatology

r cos() is the moment arm; and U is the absolute velocity in the direction of earth's rotation:Tangential Velocity = angular acceleration x moment:so total absolute angular momentum:

Slide108

ATMS 8600 Dynamic Climatology

globally integrate:Where: = mean angular velocity of the EarthI = Moment of Inertia

Slide109

ATMS 8600 Dynamic Climatology

Separate into Earth and atmosphere parts.then:so: (1) (2) (3) (4)

Slide110

ATMS 8600 Dynamic Climatology

Like the water budget, we’ll retain terms neglected in the General Circulation, that is (1) and (2), and (3). We'll eventually assume that the external torques are negligible  0What are each of the terms?

Slide111

ATMS 8600 Dynamic Climatology

Term (2)  Is the change in the length of day. Now we know that; Re >>> Ra and Ie >>> Ia thus this term should be small. It is typically estimated as a residual!

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ATMS 8600 Dynamic Climatology

Change in length of day………… ugh, those long days!Journal of Geophysical Research: The two types of El-Niño and their impacts on the length of day, O. de Viron1,2,* andJ. O. Dickey3, Article first published online: 20 MAY 2014 DOI: 10.1002/2014GL059948 Interannual and decadal scales ENSO changes, El Niño makes the day longer by about 1 x 10-6 s Also see: http://www.aoml.noaa.gov/general/enso_faq/

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ATMS 8600 Dynamic Climatology

Term (4)  This is the momentum budget (13 terms) from Atmospheric Science 8400A total change in momentum B horiz transport C vert transport D Earth Momentum horiz E. Earth Momentum vert. F MNTN Torque G Fric Torque H. Gravity Wave Torque

Slide114

ATMS 8600 Dynamic Climatology

Terms (1) and (3) together

reflect

the mass shifts of the Earth’s core, glacial cycles,

changing

ocean

basins, tectonic activity, etc……..

One the scale of climate, the LHS of (4) goes 0! This is the good approximation we discussed in

general circulation,

and as such considered balanced

!

Consider the

angular momentum

balance of the

atmosphere

(this is what we mean

)-

Slide115

ATMS 8600 Dynamic Climatology

Recall, easterlies imply the earth is giving

momentum to the atmosphere (example: tropics

)

westerlies

give back momentum to the earth

(example: mid-latitudes)

Implies

angular momentum

is transported by the atmosphere.

How does this take place? Mean motions in tropics, by eddies in

mid-latitudes

If there were no

angular momentum transport? Then the earth's

atms

. would move in solid rotation w/ earth! (Solid body rotation).

Slide116

ATMS 8600 Dynamic Climatology

Recall Atms 8400 equation:

Slide117

ATMS 8600 Dynamic Climatology

Recall the mountain torque is only effective where there's topography, and as long as the mountains are N-S. (Again, think of this as a "form drag" term representing large scale friction!)Frictional torques dominated! Frictional stress! Recall also the gravity wave stress term! This is relatively new. In climate time-scale, we assume that the sum of the external troques are = 0, which is appropriate for these time scales. While this is not a constraint, it is assumed to be so based on present observation and theory (for most climate space scales this is good assumption).

Slide118

ATMS 8600 Dynamic Climatology

Thus, the

Earth-Atmosphere

system combined neither gains or loses momentum, it is redistributed within the system (recall in

General Circulation Theory it is assumed atmosphere

gained, lost and

transport

momentum).

Then we can balance the external toques and the transports, assessing the relative importance of each! This is a balance between all three terms. Transport balances with sources + sinks

Flux

Divergenece

= sources and sinks

Remember there is no implied cause and effect between transport and transfer (no one term forces changes in the others, only assume there must be dynamic balance)

Slide119

ATMS 8600 Dynamic Climatology

Recall in General Circulation:

meridional

winds don't contribute to this balance. (no

‘[

vv

]’

terms)

we also know that winds are strongest at

the

tropopause

, so the bulk of the transports take place aloft. Transfers take place at

sfc

., thus we invoke vertical transports.

In the vertical over climate time scales, vertical

transports

as done by Hadley and

Ferrel

Cells are important.

Both

horizontal and

vertical

transports

are broken down into the mean, standing, and transient components.

Slide120

ATMS 8600 Dynamic Climatology

Over climate scales vertical transports:

For transients and

standing eddies:

transports over time are nearly zero, due to continuity. The standing eddies are least known and tough to measure!

By implication then, only the mean motions can take care of the vertical transports on time scales of climate. Thus the mean motion vertical

transport dominates

, because the change in moment arm (no longer assume

Earth’s radius is constant)

.

Slide121

ATMS 8600 Dynamic Climatology

Then, Hadley cell transports

momentum

upward

eddies transport

poleward

, and

Ferrell

cell transports downward.

Since we can't calculate vertical mass flux directly from

observations (

too tough to measure vertical motion in a

hydrostatic atmosphere),

then we can use equation to calculate that quantity based on balance requirements

Slide122

ATMS 8600 Dynamic Climatology

Transport of Angular

Momentum (summarized):

At this point, identified

two

process (pressure

torque

and frictional drag) though

which angular

momentum transferred between earth and

atmosphere,

and balanced this against the required transport.

We found that frictional drag appears to be more

important than pressure

torque (

though

these are not insignificant).

Observed transport agrees rather well with that demanded for balance.

Slide123

ATMS 8600 Dynamic Climatology

You can distinguish between transport due to mean motions, and that due to

transients

and eddies. If the former dominates, then zonally

symmetric circulations

are all that is needed,

if

the latter is needed then the situation is more complex.

Vertical transports are dominated by mean motion terms, the zonal transports on gen circ. time scales are dominated by eddy motions.

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ATMS 8600 Dynamic Climatology

Heat and Energy Balance of the Climate System

The climate system as a whole, just as the atmospheric component

(General Circulation)

must be consistent with the

1

st

law of thermodynamics (internal

energy can be altered by doing work or adding heat), and must be conserved!!

We can conceive of a climatic balance of heat/energy analogous to the mass and momentum equations. We'll have to consider sources and sinks for the addition of heat. and energy and examine required balance transports. We will apply the framework to

the atmosphere - oceans -

and

cryosphere

and the exchanges between them.

Slide125

ATMS 8600 Dynamic Climatology

Fundamental thermodynamic equation: (1) (2) (3) (4) where: e is internal energy (we could use “enthalpy” thermodynamic potential – internal energy + pressure work). The derivative is shown in flux form.

Slide126

ATMS 8600 Dynamic Climatology

term (1

)

is the

“pressure work” term

term (2)

dissipation (friction)

term (3)

radioactivity (Uranium, etc

.),

chemical, geothermal

term (4)

diabatic

heating

Slide127

ATMS 8600 Dynamic Climatology

Diabatic heating……..Hrad  solar, earth radiation, space radiation?Hsens  Conduction and convection, where in the Atmosphere of course convection is dominant. Hlat  phase changes, but in particular water (here on Earth)

Slide128

ATMS 8600 Dynamic Climatology

Internal energy per unit mass: andExpressions for e:Component Internal Energy Phase ChangesAtms (a) vapor freezing

Slide129

ATMS 8600 Dynamic Climatology

Component Internal Energy Phase ChangesOcean (w,i) Sea iceLand (l) Freezing of ground and/or lakes

Slide130

ATMS 8600 Dynamic Climatology

Expression for “baseline enthalpy”: Why? We are arbitrarily but reasonably choosing 0 C as the level of ‘zero’ enthalpy, or zero latent heat of vaporization. This will imply that the oceans transport NO LHRv, and that the transport is done by the atmosphere only. If we chose 100 C as the level of zero enthalpy then, oceans do all the transport and atmosphere does little to transport LHR (only in clouds).

Slide131

ATMS 8600 Dynamic Climatology

We can decompose: vapor iceThese are Fluxes of Latent heat of vaporization and ice which can be decomposed into:

Slide132

ATMS 8600 Dynamic Climatology

We can rewrite “the pressure work” term and “dissipation” in terms of mechanical energy (KE + PE) from N-S equations:Equation of motion, and of course dot with V:

Slide133

ATMS 8600 Dynamic Climatology

Then add to 1st law of thermodynamics: To get Bernoulli’s equation: (Recall from Atmospheric Dynamics)

Slide134

ATMS 8600 Dynamic Climatology

Now we have an equation for dry static energy (Bernoulli's equation

):

Where we can set the energy terms as

: (“

zh

” I’ve run out of Greek letters!)

Җ = e + KE +

gz

Let's get ready to integrate, and put

equation

in flux

form!

Slide135

ATMS 8600 Dynamic Climatology

Here it is: Now we're applying the equations to the 3 vertical domains represented schematically in your figure:a - atmosphere column to surface of land, ice, or waterb - region below surface where seasonal changes are importantc - region in which seasonal changes are unimportant(On climatic scale – region c is only relevant for the Ocean and Cryosphere. On land these are unimportant.

Slide136

ATMS 8600 Dynamic Climatology

Remember This?

Slide137

ATMS 8600 Dynamic Climatology

On diagram:

S - defined as the surface, the

atmosphere

-

land/ocean

ice interface.

D - level where seasonal changes vanish

B - is a reference ' bottom ' level. Take B as the ocean or ice sheet bottom and assume B = D for Land

Vertically integrate equation with arbitrary limits z1

Z2

Slide138

ATMS 8600 Dynamic Climatology

Bottom boundary (Atmosphere): Integrate:

Slide139

ATMS 8600 Dynamic Climatology

Now reverse the order of integration and differentiation to get a monster!!! (We'll take a closer look at the Monster!)where lat = latent heat and diab = all other diabatic heating.

Slide140

ATMS 8600 Dynamic Climatology

Let’s re-define total energy: and the horizontal fluxes: synoptic-scale subsynoptic fluxes fluxeswhat do these mean in the oceans? Cryosphere?

Slide141

ATMS 8600 Dynamic Climatology

Also, using continuity, we can redefine two of the terms above: Next we shall assume: Z1 and Z2 are functions of x, and t: Z(x,t)And use the terminology: (sensible and latent or diabatic heating):

Slide142

ATMS 8600 Dynamic Climatology

And: For z1 = Zs, note from our earlier analysis of water vapor budget that the first term is Evap in LHR component, and second term is SnowmeltIn the above expression:H1 = Shortwave in (solar, space)H2 = Shortwave outH3 = Sensible heating (conduction and convection)

Slide143

ATMS 8600 Dynamic Climatology

Also, using continuity, we can redefine two of the terms above:Next we shall assume: Z1 and Z2 are functions of x, and t: Z(x,t)And use the terminology: (sensible and latent or diabatic heating):

Slide144

ATMS 8600 Dynamic Climatology

For z1 = Zs, note from our earlier analysis of water vapor budget that the first term is Evaporation in LHR component, and second term is SnowmeltIn the above expression:H1 = Shortwave inH2 = Shortwave outH3 = Sensible heating (conduction and convection)

Slide145

ATMS 8600 Dynamic Climatology

Then we must go through the

Energy equation

for each vertical level, a , b, and c

. Also, we must define the short and long wave radiation.

Define levels of integration as well as take care of proper boundary conditions.

Slide146

ATMS 8600 Dynamic Climatology

a) the atmosphere:z2 = zt = top of the atmosphere: (where ever that is!).assume B.C.'s for Top: (there is no topography up there) and the underlying surface (interface w/underlying land, water, ice) z1 = zs:

Slide147

ATMS 8600 Dynamic Climatology

and underlying surface (interface w/underlying land, water, ice) z1 = zs:b) subsurface "boundary layer" region where seasonal changes are important(2 m deep typically on land) z2 = zs (same boundary condition as for ATMS.)

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ATMS 8600 Dynamic Climatology

c) deep ocean and cryosphere (seasonal cycle unimportant) 200 m for ocean and 10 m for glaciated regions.z1 = zB z2 = zD(this accounts, for example, for ocean floor topography)

Slide149

ATMS 8600 Dynamic Climatology

OK let's go ahead for atms:Continue:N  approaches 0 on the scale of climate, but not so for daily values or for 1000’s of years!

Slide150

ATMS 8600 Dynamic Climatology

net radiation from the earth systemRemember! Sun does not heat the atmosphere primarily, it heats the ground which in turn heats the atmosphere!!!So……. Let’s see the equation…………..

Slide151

ATMS 8600 Dynamic Climatology

And recall:KE fairly small PE + IE  large (TPE from mixing ratio mixing ratio Gen circ)Maybe not for Individual systems?

Slide152

ATMS 8600 Dynamic Climatology

We want to write in a more useful form:Of course use hydrostatic balance, ideal gas law, yada yada yada: If zs = 0, as in Gen. Circ., we recapture Margules' theorem from Gen. Circ. which says that PE and IE are proportional (2/5), and thus increase and decrease together. We'll consider as TPE as he did. PE = F and IE = cvT

Slide153

ATMS 8600 Dynamic Climatology

PE + IE = TPE = rcpT  Now put back into our equation for energy:    Total Energy In the Atmospheric Column!!!!  

Slide154

ATMS 8600 Dynamic Climatology

Horizontal transports (as shown in Atms 8400 – Gen Circ):Horizontal fluxes of: Dry static energy Latent Heats

Slide155

ATMS 8600 Dynamic Climatology

On to Layer B!!!Where the seasonal cycle is important:

Slide156

ATMS 8600 Dynamic Climatology

Heat equation in seasonal layer: And:  latent heat of ice What happened for Latent Heat of Vapor?

Slide157

ATMS 8600 Dynamic Climatology

Also;     Transports:   and;   should be approximately constant, w/depth. We know this to be true for the oceans. But the land is incompressible, just as the ocean is.

Slide158

ATMS 8600 Dynamic Climatology

So the above is:then:The above should be approximately constant, this the expression can come out of the integral

Slide159

ATMS 8600 Dynamic Climatology

Also from continuity:This means vertical term that the gz (PE), p , and rw terms go out, leaving: 

Slide160

ATMS 8600 Dynamic Climatology

Where:Remember: for most applications (except the oceans); KE is small.Horizontal transports of sensible heating (h(3) in transport term) is small and usually neglected. 

Slide161

ATMS 8600 Dynamic Climatology

You can substitute for Pi using: Lfqiand then substitute continuity for a glacier (fundamental equation of glaciology):

Slide162

ATMS 8600 Dynamic Climatology

This cancels out Lf qi and Lf Ji from the previous expressions and allows us to express. latent heat as: to account for phase changes due to ice and snow. We already have evap at sfc!

Slide163

ATMS 8600 Dynamic Climatology

Then finally we get: Which, in the long term = 0!!!! (our fundamental requirement for balance!!!).

Slide164

ATMS 8600 Dynamic Climatology

The equation is for an arbitrary depth, this is:   We can determine the interface condition between the atmosphere and sub surface. We can do this taking the limit as: This means all integrals go to 0, since delta (or layer thickness) approaches 0. (Zs-Zd) --> 0

Slide165

ATMS 8600 Dynamic Climatology

Then you are left with (heat fluxes in = heat fluxes out):In a more practical sense: we really want to take the limit:  where e is the depth of no fluxes of radiative energy. We're playing fast and loose here, but this simplifies the situation. Good assumption for land where radiative fluxes go to zero within centimeters of the sfc.. Not as good in the oceans where this depth is 10 - 20 m, and we neglect quite a bit of water mass. HD2 and HD1 disappear.

Slide166

ATMS 8600 Dynamic Climatology

This defines an "active layer" and energy balance becomes:where HD is the total flux on the lower bound (just conduction and convection now, especially for land, just conduction). This is a statement of energy balance in this layer first published by the famous USSR climatologist Mikhail Ivanovich Budyko in the 1950's. You see this in Neil’s and Bo’s classes.

Slide167

ATMS 8600 Dynamic Climatology

When we look at models:

 

Most climate models either parameterize or ignore heat loss or gain at HD, which (oceans, most of the surface), results in misleading results. GCMS's still have yet to deal effectively with this problem! (Don't have 'active' subsurface layers over land.) GCM’s coupled to OGCMs and/or parameterize active ocean layer above the

thermocline

.

 

 

 

The equation above is a heat balance equation for the surface:

 

 

How is balance maintained?

 

How can we measure the terms?

We'll examine in more detail later!!!

Slide168

ATMS 8600 Dynamic Climatology

Layer C (where seasonal changes become unimportant!)

We can skip the derivation and apply all we did to layer B!

 

Where HD and HB are only conduction and convection! HB represents Geothermal heat flux. The major source of this is decay of Uranium, via Alpha and Beta decay, to Lead.

On scales of climate, we can ignore, but on long term (millions of years), we cannot ignore!

Cretacious

was 7 - 10 F warmer than today. Geothermal fluxes were part of that!!!

Slide169

ATMS 8600 Dynamic Climatology

The equation:

Slide170

ATMS 8600 Dynamic Climatology

We can group equations:ATMS: Subsurf:  Sub-surface all change slow with time!

Slide171

ATMS 8600 Dynamic Climatology

We know the atmosphere changes quickly, that it’s essentially a slave to the ocean, ice, and land mass changes on all time scales greater than 3 wks to 1 month!

 

All these things must be reasonably accounted for in order to understand climate and climate changes. Thus, our weather is the result of these things. This is the crux of my skepticism with global warming.

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ATMS 8600 Dynamic Climatology

Now we take a closer look at the processes represented by the equations, with an eye toward the sources / sinks of heat / energy and the required balancing transports.

Slide173

ATMS 8600 Dynamic Climatology

Radiation and energy balance terms

 

Review: Ht1 (Short wave in) and Ht2 (long wave out) at the top of the atmosphere!

Again Ht1 + Ht2 = 0 for certain scales!

 

Warming when SW in > LW out (

albedo

increases)

Cooling when SW in < LW out.

Slide174

ATMS 8600 Dynamic Climatology

Hs1 and Hs2 are surface radiation terms.

Hs3 (convective and conductive heating, sensible) (down if warm air over cold surface)

 

Hs4 (latent heating down for dew frost, etc. up for evaporation)

Hs3 + Hs4 approx. 0!

HB is subsurface sensible heating (radioactivity, etc.)

Detailed treatments of

radiative

transfer theory (the crux of climate theories!) have been treated in whole texts! We'll only concern ourselves with physical processes, absorption, reflection, or scattering of radiation within atmosphere or at the earth's surface.

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ATMS 8600 Dynamic Climatology

Shortwave

Must distinguish between the effects clear-sky radiative, and scattered radiative (when clouds present).

Consider the following:

r - reflectivity of the atmosphere

rn

- reflectivity of the clouds

Slide176

ATMS 8600 Dynamic Climatology

rt

- reflectivity above clouds

rs

- surface reflectivity (

albedo

!)

X Opacity /

absorptivity

of atmosphere (up and down)

Ht(1) - SW in is equal to R!

 

Slide177

ATMS 8600 Dynamic Climatology

Schematic (clear sky):

Slide178

ATMS 8600 Dynamic Climatology

Cloudy Sky Diagram:

Slide179

ATMS 8600 Dynamic Climatology

If clear skies, then: is the CO-Albedo!

Slide180

ATMS 8600 Dynamic Climatology

If cloudy skies, then: Now consider some of the parameters involved in Ht and Hs and what the typical values might be.

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ATMS 8600 Dynamic Climatology

For simplicity, we will consider a homogeneous earth (

zonally

symmetric) or at least that can be represented as zonal averages (most early models represented climate and '

rt

' principles in terms of zonal averages.)

R = R(

f

,t

)

solar radiation on a horizontal surface at the top of the atmosphere.

 

X = X(

f

,t

)

shortwave

absorptivity

, which is a function of zenith angle, which determines a path length.

r = r(

f

,t

)

shortwave clear sky reflectivity

 

r =

rs

(

f

)

surface reflectivity (

albedo

) - in reality we know this varies widely w/r/t longitude (land ocean differences), and also with time on very long time scales.

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ATMS 8600 Dynamic Climatology

rt

=

rt

(

f

)

shortwave clear sky above cloud reflectivity (little dependence on longitude)

 

Xn

absorption due to cloud droplets (assume fairly uniform everywhere, but we know that may not be case for each type of cloud) (however, you cloud treat as water and ice clouds?)

rn

=

rn

(

f

)

cloudy reflectivity (which in reality is a function of cloud height and type).

Let's expand the

absorptivity

X as follows:

 

X = XCO

2

+ XO

2

+ XO

3

+ X

aerosols

+ X

vapor

Slide183

ATMS 8600 Dynamic Climatology

First three terms - fairly uniform spatially except for Antarctica in the spring

 

spatially X

areosol

may not be spatially uniform but pretty much so on large-scales (except for

pullutants

and volcanic activity).

Xvapor

= not spatially uniform at all.

 

Group the first four together as

absorptivity

for dry air, Xv is of course moisture

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ATMS 8600 Dynamic Climatology

In order to make some calculations of albedo and absorptivity, we must make some simplifying assumptions, as in:Absorptivity is a function of zenith angle, zenith angle is latitudinally dependent so, assume:  Xd(0o)  value of absorptivity at the equator P(f)  path length as a function of latitudeP(f) = R(0) at equinox / R(f) at equinox or “normalization factor”.

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ATMS 8600 Dynamic Climatology

Where R(0) is the equatorial value and R is a function of latitude, thus P = 1 at latitude 0o and P = 2 at Poles. R varies as sin(f), then R(0) / R poles sin(30o) = 2. Then P varies as cosecant(f), or secant of zenith angle!We can also write absorptivity as a function of Water Vapor content (high correlation between Vapor and Temperature: 

Slide186

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Then, based on empirical observations we make calculations of R.

 

 

X

d

= 0.06 (dry) X

v

= 0.11 (vapor) X = 0.17 (total)

X

n

= 0.04 (clouds)

 

r = 0.14 (avg. earth

albedo

)

r

t

= 0.05 (above clouds)

r

n

= 0.42 (clouds)

The real atmosphere is of course neither completely clear nor cloudy. So the fraction of the sky n, covered by clouds is very important!

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So,Short wave at sfc.: Part 1: clear sky part of the diagram Part 2: cloudy sky portion of the diagram

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Key parameters are: (1-X-r) = m clear (clear sky transmissivity!, what gets through to sfc. and (1 - X - Xn - rt - rn) = m cloudy, the cloudy sky transmissivity  then: where the bracketed quantity is: m total

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The actual values of

m

clear,

m

cloudy, and

m

total are not well known!

 

Many studies approximate as;

m

total = (1-X) (1-r)

 

this is similar to the more complex expression, while others have used the following:

 

m

=

m

clear

(

f

)(1 - f(n))

 

where the clear sky

transmissivity

is

latitudinally

depentent

since X is

latitudinally

dependent.

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where f(n) is the % of clear sky, where f(n) represents empirically deriven cloud cover!   Clear and cloudy sky combined!  And this becomes: 

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Wait for it.

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where the { stuff} = 1-A (planetary

Albedo

approx. 0.29 to 0.35)

What % is reflected by the earth

atms

. system!!

So: Ht(1) = R(1-A) or the basic expression we are familiar with!

Many studies say that equilibrium temp of earth is T = 255 K without

atms

. That assumes

albedo

is 0. Which it is not. Consider earth albedo.

 

Short wave at top and at

sfc

are calculated, let's move on to LW out at top and at

sfc

!! 

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H

t

2

and H

2

s

-

longwave

or terrestrial radiation.

 

 

Now we begin to discuss the fundamental greenhouse effect (or more precisely, the "atmosphere" effect.

This "effect" is due to the absorption and re-emission of terrestrial radiation by the atmosphere, a portion of which is "redirected" back downward!

Most gasses do not absorb shortwave, but all absorb and emit

longwave

radiation, we won't discuss the microphysics of this (

vibrational

, and rotational absorption bands, which can "broaden" due to pressure of temp effects)

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Recall from R-T, that emissivity (

e

) =

absorptivity

(a)

Kirchoff's

Law: substances which are good absorbers are good

emmitters

as well!

If earth had NO atmosphere, it would heat up very quickly by day and cool quickly by night. The moon is a good example.

 

 

Our standard parameter is

e

or 'emissivity' . Recall that all things emit radiation proportional to T to the 4th power, Stefan Boltzmann's law. 

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Since we are interested in the overall theoretical framework of climate, we do not give a detailed

radiative

transfer treatment. Instead we bring in the "greenhouse effect" in a simplified way to view variables (

e

)'s.

Absorption and re-emission of LW radiation is dependent on "path length" of gasses which is dependent on atmospheric density.

For example, Venus has a very dense

atms

, thus it is likely that much LW will be retained. LW out < < SW in. Venus's

atms

. is composed of CO

2

(96% - and 3% N

2

) and is "very dense". Venus's upper

atms

is like ours, while at the

sfc

, the pressure is 95 bars! This pressure on earth is found at 950m under the ocean. Temps are 500 C or 900 F constantly, or relatively little diurnal variation.

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Conversely the atms. path length on Mars is very long, and the atms. is very thin here. LW out >> SW in. Mars is a good example because atms 95% CO2, like Venus, but Temps are lower. Avg. Tsfc = -50C or -63 F. however at equator, daytime T’s near 0 F and nearly -100 F at night. There is a large diurnal temperature swing due to low pressure ( 8 – 12 hPa) at the surface. This is similar to our atmosphere 40 km or 24 mi up!   The value of the atmospheric emissivity in the infrared is roughly e = 0.77. Emissivity can be different for different spectra. Emisssivity is low in the shortwave. We only “see” earth from space due to reflected SW radiation.

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The emissivity of 0.77 is obtained from linear contributions from the constituent gasses.

the "overlap" component is due to overlap between water vapor and carbon dioxide absorption bands.

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Note:

e

v

= f(T

s

) (since water vapor is strongly correlated with surf. temp.)

Atms

. "holds" more water vapor as an exponential (

Claussius

-

Clapeyron

equation).

 

 

 

and

 

e

CO

2

= b1

ln

(CO

2

) + b2 emissivity will increase as a log of increase of CO

2

which means at some point you'll reach "saturation". (Oglesby and Saltzman, 1990, J.

Clim

.). Another reason for skepticism….

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Thus, for 'man made' greenhouse effect' there is a point where inc. in CO

2

doesn't matter anymore! The real issue is inc. in the more potent (30 time more potent, molecule for molecule) greenhouse gas: H

2

Ovapor.

 

For present-day concentrations of CO

2

, about 395

ppm

, H

2

O

vapor

,and O

3

:

  

e

= 0.64 + 0.19 - 0.12 +0.06 = 0.77 (terms in same order as above).

 

For the pressure broadening factors g1, and g2, we set g2 = 0.03 and g1 = 0.00 (e.g., above cloud

atms

. is too thin for this to be important!)

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Then with the aid of the diagram, we can write the following expressions for LW radiation:

H

s

=

s

T

s

4

( 1-

) - n

n

T

n

4

(1-

)

H

t

= (1 -

)n

n

T

n

4

+ (1-n)g2

T2

4

+g1

T1

4

+ (1-n)(1-

)

s

T

s

4

a further common approximation is to assume:

e

s

=

e

n

= 1.00

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Or that the earth's surface and clouds are taken as blackbodies. (Not valid for high cirrus = 0.8 - 0.9) atms = 0.77The clear sky atmosphere remember has a considerably lower e = 0.77, this is barely a "grey" body (0.8 - 0.9).A standard empirical statement for putting the cloud tops and bottom in terms of surface Temperature: tops: bottoms:

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This last approx. assumes a more-or-less constant heights for cloud tops and bottoms, and that a standard atmospheric lapse rate applies, ie., B1 < B2.Similarly, we assume:Upper Layer: Bottom Layer: So finally:Long wave:

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Long wave: The bracketed quantity, (1-stuff), the “stuff” is what is “trapped”, and 1 – stuff is what escapes to space.these (along with the HSW) form the basic radiative parameterizations for simple RBM models and more complex GCMs.

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Let's move along to sensible heating.

We've looked at radiative processes briefly. Obviously, we can treat these in more detail in

Atms

physics.

Let's look at non-radiative heat transfer or

diabatic

heating.

Sensible heating, Hs3, represents primarily convection in the atmosphere (or the convective heating and upward rotation of the air parcels due to heat transfer from the underlying surface: "bulk transfer").

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Even though convection is the fundamental or dominant process of heating for the atmosphere. However, at some point, radiative transfer heats the surface and conduction takes place at

sfc

.

Also recall that there is no sensible transfer of heat out of

atms

. since sensible heating requires mass, which "vanishes" at top of the

atms

. Radiative processes do not require mass.

In order to consider sensible heating of the

atms

. from the underlying surface, it will be necessary to consider the entire extended Bound. Layer, and then partition it.

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Монин-Обухов (1946)  Similarity hypothesis, examines the lower part of the PBL as an “accordion” that stretches and contracts as a function of stability

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Monin and Obukhov also define “mixing length” or the rough size of eddies near the surface:

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We distinguish between Ta (air temp in the shelter, or standard temp. height (2m) and

Ts

, which is the temperature of the surface (air/water, air/land, or air/ice --

zs

) interface (what you feel on the asphalt on a hot summer's day).

This

is also

Tg

or temp of ground or "soil" at an infinitesimally thin layer at

zs

.

The key then is the temp difference

Ts

- Ta

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For a synoptic average, we can write from BL theory "Bulk Transfer" or Eddy Mixing (turbulent) theory: which is subsynoptic motions.we make this statement assuming that Hs3 is accounted for by convection. What we say is that (turbulent) vertical velocities carry warm air up and cool air down across the Surface Boundary Layer. However, w'T' are impossible to measure, so:

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We use standard "mixing length theory" (from BL meteorology) which assumes that the correlation w'T' is proportional to lapse rate! We can ignore adiabatic compression effects as we move across the SBL (10m), but could not for whole PBL.So: where k  is an eddy mixing coefficient.

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and:where This is the “Bulk Transfer method commonly used; e.g., Neiman and Shapiro (1993) MWR Aug

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If one "knows" D, then we don't have to compute za - zs, directly, all we need to do is know Ts and Ta. Typically, K and D are empirically derived coefficients, derived through experimentation.Now 'D' is some measure of the 'mixing magnitude' (see Arya, Introduction to Micrometeorology). From mixing length theory, D can be approximated as:

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again, from empirical data, a = 0.5 - 0.7 and b = 0.2, so typical values for D = 0.01 m/s. These derived values are for typical wind speeds. finally, we get: Which again, is recognizable as the standard bulk aerodynamic formulation.

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This formulation is generally provides good results over water (over the oceans), since the surface is smooth, but is less effective over land.This stinks because LHR is far more important over water (low Bowen ratio surface), than over land (high Bowen ratio surface).Recall Bowen ratio:

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One method in getting better results is the Budyko approach - don't measure Hs3, instead, assume balance and calculate as a residual, since even though our estimates of Hs1 and Hs2 are parameterized, we do these well. Sensible heating over land is the most difficult parameter to calculate. Climate is a BL problem, so, we'll look at BL meteorology.

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The Budyko approach is fine, but doesn't work if we want to actually calculates surface temperatures, so one way or another, we must explicitly treat.So if we want a unique soln. for Hs3, we need a relationship between Hs3 and Tg, but also more importantly a reliable way to measure Tg.Let's generalize the approach previously examined:linear decreases with height, where now we consider the extended PBL, Ts = 1000 hPa and Tz = 800 hPa.

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With the deeper layer, we account for adiabatic expansion:so:where “F” denotes the adiabatic lapse rate, and “” is a "counter gradient flux factor" (fudge factor) so that the atmosphere does not have to go fully super adiabatic (not common on large-scale anyway) for convection to occur. This is a parameterization applying small-scale theory to large-scale and including a fudge factor. Otherwise, convection would never occur in the model.

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Now we need to make some assumptions to put our expression in a more usable form: Now that we have deepened our layer, and for climatological averages of Hs3, we would like to be able to take account of the diurnal cycle (far more important for land, but relatively unimportant for the ocean), and synoptic ' pulses', which are important for land, but not for oceans.

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Well we've looked at Reynold's averaging, now let's "extend" that concept:where mm = refers to a monthly meannow, the perturbations can be broken into components: where we have a diurnal departure and a synoptic (tranisent) departure. We're taking into account both the diurnal cycle and the synoptic-scale in the variable.

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Thus for temperature T we have: and thus:

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so:This is to assume that top of EPBL (800 hPa?) there is little or not diurnal variations!

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So we assume that typically our profiles in the EPBL look like:

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Temperatures in the PBL:

1) We have small diurnal cycles in free

atms

. since

atms

. does absorb some SW in.

2) Small diurnal cycles over oceans for a) ocean has large

Cp

, and b) oceans mix heat downward.

3)

Ocean will respond slower to air temp changes and they influence

each other

.

4) Land responds quickly to T changes.

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ATMS 8600 Dynamic Climatology

To a first approximation, we can set:where  is 0.7 and is empirically determined. This says that empirically, the T change at the surface is 70% T change at 850 hPa.

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what about variances?As a first approximation, we can assume T variability is proportional to the T gradient (T gradients found along polar fronts - or the storm track). Where A is the Austauch Corefficient, which is an turbuelent eddy mixing coefficent and  is a "newtonian cooling" coefficient.

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Newtonian cooling  no heat added, heat leaves the system as described by: No good representation of the diurnal cycle temps exists, so we'll leave it for now. Thus we can write quite simply that sensible heating is composed of three terms:

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Hs3which is the newtonian cooling partition involving mean climatological temperatures, or heating and cooling by LW radiational processes and conduction.Hb is the synoptic warming/cooling and Hc the diurnal warming/coolingThis ends our look at Sensible heat flux.

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Now due to phase changes (we restrict ourselves to evap + cond). We won't worry about changes of solid to liquid here.OK: If we restrict to evaporation and condensation, then Latent heat flux is: But, how do you measure Evaporation?!!Instrumental observations :Assume:

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where w’q’ term is the vertical flux of water vapor through the SBL, with the implicit assumption that mean vertical motions and transport is zero ( = 0 or q = 0) so we can measure w’ and q’ directly??What we're measuring is the average of the "bursts' through the SBL. Thus maybe, as was stated y'day Theta surfaces are better for moisture transports. This method is good for small regions.

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We could use Hydrologic Balance requirements:This is good for large areas:for atms: for sfc:

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so if we know precipitation P and either Atms flux divergence ( ) or the runoff flux div ( ) then we have estimates for (E + Sub).This methodology is fine for measureing Hs4, say, for model ground truthing or verifications. In order to deduce Hs4 (for a climate model) we must write (like sensible heat) in terms of model computable quantities.

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Thus as with Hs3 and going back to the eddy mixing formulation and making use of turbulent theory as in Hs3 we write:Where D is the drag coefficient

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and we know that mixrat or specific humid is: where “e” is the vapor pressure, and saturation specific humidity is: With as shown by the Claussius- Clapeyron Equation and RH is:

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And, continuedthen

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A frequent and very critical assumption that is made by all climate models is: or that Drag. Coefficient for H20Vapor = Drag coefficient for sensible heating (some equate momentum drag coefficient as well). Thus these are lumped as one constant which assume that heat and vapor fluxes are subject to same turbulent mixing. Then as for sens. heat:

Slide236

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Then This is the bulk aerodynamic formulation for transpoort of LH. This is very appropo. for the oceans where vapor pressure nearly equals sat. vaopr pressure. But, land is a different story again!!

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Then to begin we note for a saturated surface that: which is the potential evapo-transpiration (how much would we get if an infinite supply of H2O vapor?)Theoretically, real evaporation = potential evaporation over the oceans!

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But for land we can define a water availability function (surface saturation factor, or the ratio of actual to potential evaporation!)! D is not easy to evaluate, but now that we have a Sensible heating paramerterization, we have

Slide239

ATMS 8600 Dynamic Climatology

Then plug that into equation for Hs4: then manipulate and get a ratio of sens to LH, which is defined as the Bowen ratio:

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What is Hs3 assuming 1/B is known?One method (use heat balance at sfc):Now if we assume Hs3 balances over an averaging period of interest, then LH in terms of radiative fluxes:or Hs5  0

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But we still need (a model friendly) B:One still very good method of evaluating B is classic “Penman“ approach: multiply strategically by 1 (saturated e) :

Slide242

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which yields: the reason for this is, rewrite finite difference as differential: which of course can be calculated from the Clausius-Clapyron Eqn. and this quantity is hence known!

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Since we can make use of alegebraic expressions (A) and substitution (B):The substitute and do some alegebra:

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Solving for Hs4:Now all quantities here are know or at least measureable!If we assume that Hs4 is a saturated point,

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Since we already have an expression to deduce or calculated Hs3 over land which we can cluclate in the model, we can now deduce in the model Hs4 as well.!!!Subsurface Heat flux (Hs5):For land, heat conduction is the relevant, but very slow, process, while for the ocean, convection is the much faster and more relevant process.We write:

Slide246

ATMS 8600 Dynamic Climatology

Kt

- thermal conductivity (not

cretacious

-tertiary bound.)

Kt

=

CpK

(where K is thermal diffusivity, which you can look up in a CRC)

over land, no convection (parcels don't move up and down), ocean has heat conduction too but heat conduction <<<<<< convection.

Oceans great storehouse of heat 1) convective and mech. mixing. and 2) larger heat cap.

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then Hs5:Now the tables are turned, ok for land not for oceans!For ocean: where Td is the temp at the thermocline (seasonal changes go to 0, or oceanic equiv of tropopause)

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Now for oceans K is not constant, it's a function of Temp. and salinity K(S,T). We could write:where Ko represents strictly mechanical or wind mixing. Note that this gives a relationship as a function of Td and Ts, and Td is typically the temp of the coolest winter month, since wind or mechanincal mixing is greatest in that month!

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Does Energy Balance and how?

Spent several weeks deriving energy equations, and then looking in some detail at heat fluxes at the top on the Atmos. and at

sfc

.

Ok let's return to our energy equations and balance sources against transports, quick review of 416.

Energy equation:

Ea

+

Eb

+

Ec

(all layers,

atms

, and surf. and

subsurf

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And away we go.

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To compare to observations, we will need to specialize for atmosphere only.where

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Recall in gen circ. we considered a volume integral. And then we've done a 30yr climatological time average:Again this is our fundamental requirement of a climatological steady state. In such a state, time derivative is zero! Thus, the sources and sinks must be balanced by transports.

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The last term on the right, is the energy source term, fundamentally. Of course the Solar radiation drives climate and hence is the source of energy. (it is also convenient to consider HT2 (long wave out) as part of the energy source for the climate system). When HT1 and HT2 balance at every point on the globe, there is no corresponding transport, since radiative balance would be maintained at any point. If they don't balance on the globe (as they do not) then there is an implied horizontal transports.

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We're familiar with energy balance and implied transports:

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These transports are done by the mean motions, standing , and transient eddies: No heat transports of any consequences in the land or ice surfaces, but there are in the oceans, which we can solve as a residual?

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Simple theoretical Expressions for the Observed Climate!We'll begin by examing the thermodynamic only energy balance model. Recall for any point on the earth's surface: where E is total energy of a vertical column per unit area and F is vertically integrated horiz. flux of energy. Initially consider annual and global averages!

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If we work within the fundamental climatic requirement, or energy balance constraint: We would like to be able to relate this expression to surface temperature Ts, so we're going to have to solve for that quantity:

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Recall how we evaluated Ht (SW in):(we can write having applied a time and zonal mean): where A is the (zonally averaged) top of the atmosphere (planetary) albedo: thus the co-albedo or planetary absorptivity is:

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We can also write: where So is the solar parameter, ~1380 W/m2 (amount of SW impinging on a plane at top of atms.For simplicity define:

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then a global average for Ht1 is: where Ap and ap are now the global planetary albedo and co-albedo respectively. We will go back to these quantities being latitudinally dependent later.ALL EBMs and GCMs have a statement like this whether it is empirical or analytical!

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Now consider HT2:The simplest thing to do is to set:or we assume that the earth atmosphere system cools as a black body at an "effective' temperature:(T skin < Teff < Tsurf) Teff is a well defined physical quantity, but can't readily measure it.

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Thus: so: Now with Q  340 W/m2 and assuming a present day value of Ap  0.3, Teff  255 K

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This is only valid for present atmosphere composition and albedo. This is NOT the oft quoted temperature earth would have without an atmosphere since,

Ap

would no longer be 0.3. Then, the effective temperature would be: 271 K (Albedo closer to 0.1, similar to moon!)

This can be considered the 'effective' temperature of the earth-atmosphere. system. This temperature is reasonably close to the triple point of water, thus in essence incorporating a fundamental zero-order climate theory: earth is very close to triple point of water, this is a very important and powerful influence on our climate.

This is an interesting, but not very useful result, but can't we do more?

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Budyko (Будыко) 1969 Tellus 21 611 - 619a and b are empirically determined from the data:

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these empirically take into account the global effects of green house gasses, and H20 and CO2. Budyko found these based on real data and independent of temperature. Most important greenhouse gas is water vapor.Then equating Ht1 and Ht2 (we now have a very simple EBM!)

Slide266

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We get

Ts

= 285 K for current

Ap

of 0.3. (Which is reasonably close to the observed value of ~288 K for Earth's temp.) Better parameterizations exist today, I'm sure.

Now, what if

istead

of holding the Albedo constant, we made this a function of surface temperature? A = A(

Ts

), this then allows for ice-albedo feed backs. Thus, this paper of

Budyko's

got the ball rolling on climatology. Explain what I did in my experiment!

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Here's the functional representation:

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so now: and we have 2 equilibria defined by intersections of I, II, III. Note state II is unstable, while state I and III are stable. I = ice covered earth, and III is little or no ice (present day case)

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If Q is reduced, then ice-covered equilibria may be possible,

backsolve

for Q with a -15 C temp!

If Q is increased one can expect to reside at little or no-ice conditions.

Thus, ice ages represent unstable climatic modes!

when he performed his experiments, he obtained the

phase portrait from earlier

(in his "planet

Budyko

" model!!!).

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If you move down the stable present-day, and lower So enough - you can jump to ice-covered state. If you move up the ice covered state, you can jump to little or no ice. Or you can follow the "unstable" path. You can stay in stable states also, but on unstable branch you must move or jump toward one or the other.

This is common behavior for non-linear (chaotic) dynamical systems!

since one can jump from one stable regime to the other, this implies that small changes is Solar parameter (S) (a few % depending on it's exact value) will cause a drastic change in

Ts

!

Now, does one put more or less credibility in "solar variations" as a cause of current climate change? The constant "

diss

" of solar variations is that the % changes are too small!!!

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Climate Sensitivity (change in quantity A vs. B)Let us now attempt to formalize the sensitivity of Ts to changes in the Solar Parameter (obtain a global measure of climatic sensitivity) or other radiative forcings.Then we define sensitivity (S):

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or the change in

Ts

per a 1% (variable “x”) change in Q. We could pick some other level,

ie

5% change in Q. Then Divide by 20.

What kind of change in the Solar Parameter? 1% * 1380 approx. 14 W/m

2

. 5% change would be 70

W/m

IPCC

Definition

dTs

=

*

RF. (Here RF = B/x)

And

dA

/ dB can be estimated

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Eschenbach

, Held and others

T1 = To +

* RF (1-a) +

To *a

Where the equation above is a modified linear. This attempts to put a non-linearity into sensitivity to replicate ‘feedback’

a =

e

(-

1/

)

where



is a ‘lag’ time constant. This is similar to “Newtonian cooling” in concept.

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So, define:then, and take partial w/r/t surface Temperature,

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after substituting (2) into (1): and now we can used Budyko's or any other expression for or take the partial w/r/t to surface temperature again,

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since b is a positive number in the denominator (or a measure of the strength of atmospheric greenhouse), and the larger it is, the less sensitive the climatic equilibrium is.On the other hand, since is subtracted from b, the larger the ice albedo feedback, the more sensitive the climatic equilibrium is.If, for example, , then S = 0.63 K, or a 1% linear change in the solar constant yields a 0.63 K linear change in Ts, if we consider the Earth atmosphere system a black body.

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This change in temperature is roughly that experienced over the course of the last century. Thus, we have to ask, has the Solar Parameter changed by 14 W/m

2

? Or does solar variability explain the change in climate?

The answer is not likely, even though the error in measurement of So is large (1368 - 1380 depending on whose text!). If we consider earth a black body, then system is not very sensitive.

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Let's try Budyko's parameterization:Use , with a = 203 W/m2 and b = 2.09 W/m2 K, then B = 1.12 K. So a change in So of 14 W/m2 leads to a change in global surface temperature of 1.12 K, or Earth atms. system is quite senstive to changes is So 7 W/m2 would produce 0.55 K. Solar variability can be quite influential in climate. These are also linear changes and thus when we "jump" from one climate state to another, we can toss this out the window!

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Is solar variability responsible for some of our noted rise in the last century? Yes, some of this can be explained by solar variability. Some estimates of Solar variability are near 4 W/m2, thus, as much as half the "Global Warming" might be solar variability.As a final note, we can write: and let:

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and again: and substitute in, Now we have a simple diff equation and a (-) or exponential declining represents damping, or + sign represents exponential growth. We also have a battle between the two terms in the second term on the RHS of the equation.

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b is the longwave radiation which dampens sensitivities. , the albedo feedback tends to destabilize the system or enhances sensitivity.A 1 - D zonally averaged EBM (How to build a model!)This is a simple model. For climate this can resolve latitude variations but zonal and vertical avgs are imposed. a 2-D model would resolve lat-long. and represent vert. integrals. a 3-D model would resolve all three dimensions. 0-D all global averages (Budyko model we just finished).

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The basic energy equation: flux term (zonal averages)unlike the 0 - D equations of Budyko's we now have a horizontal transport to deal with.Now: specify or parameterize:

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Now: specify or parameterize:1) short wave radiationf() is a well-known and tabulated function (as one might guess, it is related to mean solar zenith angle).What about A()? Since, as with global EBMs, we will want to consider albedo feedbacks, we will again consider piece-wise functions in latitude this time, though of a different sort then before.

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In other words, a step function at the permanent ice latitude.How how do we determine i? We can define it in terms of the latitude where a particular value of Ts occurs, thus making i a variable computed by the model!

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So: where i is the (poleward) latitude where Ts < = -10o C first occurs. This is a reasonable, but somewhat arbitrary assumption as to where permanent glaciation occurs.This will be the feedback mechanism in the 1-D zonal EBM!!!!

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2) Long wave radiation:As before we'll use Budyko's formulation: 3) Now the biggie: how are we going to specify the heat flux or heat transport term?Since this is a simple energy balance model, we won't be using N-S equations to get or anything like that ( ). We'll have to make more assumptions.

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Budyko and Sellers in 1968 came up with independently derived parameterizations:Budyko's: Zonal Global Avg Avg T’s T’sThis is a Newtonian cooling type of formulation with the heat transport and any latitude being proportional to the temperature difference between the value at your locale and a system (global) average.  is an empirically derived constant, which is prescribed.

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Sellers: so that the heat flux (transport) is proportional to the surface temperature gradient at any latitude.again, K is an empirical proportionality constant that must be prescribed.(generally prescribe by running the model and seeing which value gives right answer! The old fudge factor!!!!)

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Now plug into Energy differential equation and here is your simple model:Simple Budyko energy balance model!Recall from the 0 - Dimensional global model EBM we write:

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Substituting and solving for Q yields (why solve for Q the external forcing? We can see how sensitive our climate model is to solar forcing. In 1970, CO2 problem was not an issue):So we can in fact solve for the value of the Solar parameter required to place fi at any desired latitude.

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That is: using Ts(i) = -10 C, Ap(i) = 0.47 and ap(i) = 0.53and a and b are as before.

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Thus what we are doing is what we call solving for the "inverse problem". The inverse problem means solving for the external forcing that gives a certain condition. Used in Remote Sensing applications (here’s radiance, what’s the temperature?). Analogous to Jeopardy, here's the answer, what's the question?

Look at specific cases:

First, consider the case where gamma = 0 (or no heat fluxes), local thermodynamic balance

onl

we see a monotonic change from

i

= 0 to

i

= 90 as S/So increases!!

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Now consider the case gamma not equal to 0

We get a similar looking curve to that for the global model, the "push" - "pull" type response. To push the solar constant beyond what it is today to force chances.

The point is: Climate is very sensitive to small changes in So. (This model may be too

sensitive

). you can "jump" from ice covered to ice free or some intermediate stage.

So we now see the same sort of "non-linear" over shooting and jumping behavior as in the O-Dimensional EBM.

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How does changing the strength of the heat flux through the empirical constant change results?Plot Ts() vs :when gamma is 0, then Ts curve is essentially that of f(), giving the latitudinal distribution of Q.when gamma goes to infinity, uniform temps are found at every latitude

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Thus, the real transport in 1/2 way between. For the EBM you must tell it what heat transport is going to be?

Inadequacies of the EBM:

1. No capability for deducing:

a. zonation of climate (Hadley and

Ferrel

cells) associated with mean

meridional

motions. Can't replicate, must specify.

b. of course no land seas zonal variations

c. hydrologic cycle

d. deep ocean states and ice

thickni

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2. F = v'T' heat flux poorly parameterized in terms of Tsa. should depend on atmospheric T not Ts, or more importantly temp gradients (hemispheric asymmetry)b. no latent heat fluxc. no energy flux due to mean motions.3. should depend on atmospheric state (clouds) as well as surface state4. Lower Boundary condition at Z=Zd (Hd) is important , but ignored in this formulation. This is especially important over the 70% of earth that's water covered.

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EBM does this well:

Computes latitudinal (zonally averaged) distributions of

Ts

as well if not better than more sophisticated GCM's. (No dynamics here, so it's simple)

This is what the model was built for and it does it well! But we want other things (this model is quite limited) as well as

Ts

!

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Introducing dynamics into a simple Climate model (on the road to a GCM).

The most important factors we now must consider putting into a simple climate model:

1. Zonation (mean

meridional

motions)

2. Water (hydrologic cycle)

Precip

and

Evap

3. heat flux dependent on

atms

. Temp and temp gradients.

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4. Non diffusive transports (counter -gradient stresses) (possible w/ momentum eqn.)

5. Clouds (This is important for radiative balance).

6. heat flux

trhough

the boundary z=

zd

or

Hd

(the thermocline in ocean,

seaonal

change layer). Take into account deep ocean and geothermal heat flux.

This is the essence of a statistical dynamical model or SDM!

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Schematically our domain again

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Unlike the EBM, we will explicitly treat a) and b) in different ways. We will have to specify Td in the ocean (i.e. specify the heat flux into the deep ocean), while on land Td is determined from annual cycle.

Thus, now we have 2 conditions, one at the top of the model (incoming and outgoing radiation) and one at the bottom of the model (Td, the temperature at the base of the seasonal thermocline)

Typically, it's that of the coldest month since mixing is effective down to a certain point.

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The task is: Given Ht1 and Ht2 and Td, determine separately equilibrium states in the atmosphere AND subsurface Boundary layer. Determine: for each layer!Some reading Saltzman and Vernker, 1971 JGR, 76 p1498 - 1524.Modification by Oglesby and Saltzman, 1990. paleogeography, paleoclim., paleoeco., 82, 237 - 259.

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One layer

atms

, 2 layers overall:

_________E in =

Eout______ey

(

out)__(1-a)x out___

Atms

gains ax but loses y out in both dir. (

ey

)

______________gains

ey

loses ex_____________

sfc

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I will solve problem for N=2 (one

atms

. layer and surf.). Also use principle that

Ein

=

Eout

or

Ein

-

Eout

=0, which means each layer is

radiatively

balanced!

 

N = 2

surface: x - y = E

atms: ax - 2y = 0

  x y E

| 1 -1 | E  | 1 -1| E

| 0.5 -2y| 0

 | 0 3 | E

Use Gauss-Jordan method to solve for x = 4/3E (surface emission), and y = E/3 (atms. emission). (mult row 2 by -2 and add to row 1)

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Also, I'll hand out a comprehensive list of the basic equations specialized for the atmosphere, ocean, and cryosphere.  Note the equations have been climatology averaged!! That is, they are expressed in terms of climatic means and variances. The critical feature is the introduction of the momentum equation.Remember we can write:  uo u1

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Where

u

o

= is the

zonally

-symmetric portion of the time mean and u

1

is

zonally

-asymmetric portion of the time mean (standing eddies). We don't have a concrete def. here for standing eddies. How do they come about? Mountains or large-scale SST's or combination force them. 

 

As it currently stands, the model solves only for

u

o

expanding the model to include standing eddies as well is a topic for future research. Again, our theoretical understanding of these topics are not nearly as well understood as mean motions.

 

The first step to developing the model is to take the time-averaged equations (see hand-out) and perform zonal average, separate out the

zonally

-symmetric and asymmetric parts. (Following Saltzman, 1978).

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When we do this, we get the following for the zonal momentum equations:

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To shorten notation: Transient Eddy Stress:   Standing Eddy Stress:

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M is the transient eddy stress terms, and N is the standing eddy stress term. Note that these terms in N are not in form of and hence not set to 0, since they reflect the effects of standing waves on the zonal average. Also, we cannot solve for symmetric circulation w/out considering the asymmetric circulations.   We need also to go through a similar procedure for the thermodynamic equation which for convenience we write in terms of potential temperature, rather than actual temperature T.

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Potential temperature:this yields: (1) (2)   (3)

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Where term (1) is the Mean motion term: this term is fundamental, if we can't calculate this, we can't get eddy term. This term is ignored by the EBM.

 

 

And term (2)

Diabatic

heating term. EBM solves for this.

 

 

Term (3) is the eddy term

use mixing length theory, these are parameterized as heat transports.

 

 

Thus, the EBM ignored term 1, parameterized term 3, and explicitly calculates term 2.

 

Now the ultimate goal of our model is to deduce, given R

o

(t) and

TD

o

, the climatic quantities:

 

u

o

,

v

o

,

w

o

,

q

o

,

q

o

, P

o

,

E

o

Where q is

precipitable

water, P

o

is precipitation and

E

o

is Evaporation. 

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Let's write down then our set of fundamental, zonally symmetric, and SEASONALLY - AVERAGED (but not vertically averaged) equations: U-eqn

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V - eqn

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Hydrostatic balance:  Thermodynamics:

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For the hydrologic cycle, we include the water vapor continuity eqn:  Note that we have switched from w to w so as to retain evaporation when vertically integrated.

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Now in the equations all the stress terms must be expressible in terms of the mean motions

u

o

,

v

o

, etc.

 

Also, in essence we do an inverse problem of obtaining small scale feature from large-scale features.

 

 

Also, the

diabatic

heating

Q

o

must be prescribed or calculated!

 

The above equations are for the atmospheric part of the system or layer a. What about the subsurface, or region b?

 

 

We will use the energy balance equation and demand that the heat fluxes at the boundary between a and b balance!!!

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Energy Balance:  Solar LW sens LHR Deep ocean heat fluxThus to proceed, we must now make some approx./ simplifications. These are the statements which will make the model!!

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 1) where d = season, could use "warm" and "cold" (6-month seasons), or 4 (1/2) "warm" and "cold" seasons, or all 4 seasons (bar is over brackets). 2) for any  our fundamental equilibrium requirement, and that's 1) and 2) go hand-in-hand. 3) neglect transport due to standing waves, e.g., for any (we have already neglected terms  now we'll neglect these as well as the effects of standing waves on the zonally symmetric circulation).

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assume that (away from the equator) the zonal wind is in geostrophic balance, that is:    which also yields for the thermal wind: 

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5) w = 0 at p = ps and p = 0 (rigid - lid assumption). 6) Treat the SBL as a separate layer: and where we assume CD = CDo u, or there is a linear dependence of the drag on u. (good for areas outside the tropics. In tropics where surface winds are weaker, a different relationship is needed!) Then assume that friction in the SBL balances the geostrophic term:

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7) Vertical Approximations and simplifications in the SDM (use standard gen. circ.

arguements

)

A)

uo

is assumed to vary linearly with height in the

atms

., thus if you know

uo

(surf) and the thermal wind, you know

uo

everywhere.

 

Over land and ice: SBL we assume

uo

goes to 0 at z(surf). Over water,

uo

not necessarily 0.

 

 

b.

vo

assume simple overturning:

 

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c.

w

o

assume 0 at z surf and

ztop

, standard large-scale and synoptic scale

arguements

from continuity, which dictates a "

cos

" wave in vertical.

 

 d.

q

assume a linear increase with height (cond. unstable), the slope of which depends on the lapse rate. 

e. assume water vapor is at a maximum at the surface (

qs

) and decreases vertically using some power law, such that the amount is negligible by 500

hPa

(non-div. level). Assume the same rate of change for all latitudes, but of course

qs

varies with latitude (large in tropics, small at poles).

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Now total mass of vapor is small above 500

hpa

(small compared to below), and is thus negligible for the hydrologic cycle (E- P), but not unimportant for

radiative

balances in upper troposphere and stratosphere (selectively ignore in this case).

 

 

f. Vertically averaged values of anything are attributed to the 500

hPa

level. The traditional level to look at in upper air, also half the mass of atmos.. typically a 500

hPa

value represents vertical average well.

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Summarize these with the following expressions;

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and finally

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for Hs is the relative humidity at the surface, assumed constant or nearly so over globe.

RH globe = RH ocean + C (typically 1 - 5%). Models are making RH an output variable as opposed to specific humidity or mix. rat.

 

Now you insert our approximations into the basic equations and get a reduced set of equations (averaged in the vertical).

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Momentum:transport of momentum by the mean and eddy motions frictional drag

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   Thermal WindEkmann Balance in PBL: geostrophy fric

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Dry Mass Continuity:Water Vapor cont (Basic budget equation or water vapor equation): Transport Source/sink

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Energy Equations:  Parameterize eddy stress in terms of mean motions transport

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We still need to parameterize the stress terms and the heat fluxes to obtain a closed system!These equations are for the atmosphere. For the subsurface layer we have simply:   The unknowns we wish to solve for are:

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We still need to consider how to parameterize the heat fluxes. Recall, all models use variants of similar forms we examined earlier. Here they are:Sub-Surface and Surface Heat fluxes: Solar:

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Where n is a cloud parameter.Long wave: atmospheric and surface radiationwhere v is an emmissivity factor. In the real atmosphere, radiation assumed isotropic – or equally in all directions+ value is downward, and – value is upward.

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Sensible heating (a newtonian formulation): sensible heatingLatent heating (where w is water availability) -what % of potential evaporation is actual?

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Subsurface heat flux: temperature at the depth of the seasonal cycle where it goes to zero.where, mechanical mixing “static stability” for water

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kl is constant for land and icekw is for waterAtmospheric:

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Then you must tabulate prescribed parameters (obtained from the data) the we need to evaluate the above atmospheric variables (all are free, you just have to prescribe them!). You can change these to fit your particular experiment. This model used to examine paleoclimate, thus we need to prescribe them!You have: the second is temperature at the base of the thermocline (from the coldest month)the atmospheric radiation parameters

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surface state parameters:where jn() = fraction of the latitude covered by surface type!

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Prescribe 5 surface types:bare land, open water, snow continental ice, and sea ice.

N

type

k

w

r

s

(

f

)

 

 

 

 

 

1

Ocean

k*-c

6

(T

s

-T

D

)

1.0

a(

f

)

2

Sea ice

5

0.5

b(

f

)

3

Land

1

0.8

0.2

4

Snow

1

0.5

b(

f

)

5

Continental ice

1

0.5

b(

f

)

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Now prescribe or parameterize the stress termsFirst, we consider the transient eddy heat transport term as with the EBM, we set: Transients (strength of term) whether there be more or stronger eddies is porportional to temp grad.

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Now, the big questions, what to do about “k”:1) Constant (simplest approximation) a prescribed Austauch mixing length coefficient. That was what was done in EBM.2) proportional to the heat flux.

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3) k= constant, but not prescribed- rather determined from some integral constraint.4) Baroclinic adjustment; the climate system adjusts the Temp. grad. to maintain constant neutral stability. This process is being developed.OK the SDM uses # 2 in this form: eddy variance of meridional wind  measure of “storminess”

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where A is a model creation horizontal gradient lapse rateEddy water vapor transport: used simple mixing length argument – considerationsthese are quite plausible in this case!

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moisture gradientsB has an implied dependence on the “storminess”.Eddy Momentum transport: u'v' based on strength of zonal wind and/or vorticitySee me for more detailed treatment of such

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A Hierarchy of Equilibrium climate models (Simple to complex)

1. Energy balance models (EBM)

2. Radiative Convective Model (RCM) - examines radiative transfer for a column of air, takes into account convection in the column.

3. Statistical Dynamical Model (SDM) solves for means or ensemble conditions.

These are all

true

climate models.

Each

will give you a result that represents the climate system are initialized with climatic data.

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4. General Circulation Models (GCM) Not a true climate model.

does not solve for climatic properties, solves for instantaneous 'weather', using "weather' theory.

That is in particular that weather forecasting is an INITAL VALUE problem. GCMS are intransitive in the short run in that result is SENSITIVE TO INTIIAL Conditions. However, they are "deterministic" in the sense that they relax toward climatology after about one week, which can only change w/ a change in BC's. Lorenz- fundamental

predicability

for about 2 weeks. Thus, they do not behave like true atmosphere for gen. circ. and climate scales.

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Order of development is different though:

1. GCM - developed by N. Phillips and

Smagorinsky

in 1950's

2. RCMs developed to put RT code into GCMs in early 1960's

3. EBM developed by

Budyko

in USSR and Sellers in US independently.

4. SDM - developed in early and mid-1970's

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The EBM:Computes system or surface temperature by balancing of incoming and outgoing radiation.Essentially valid for time scale of weeks or longer, thus diffusion averaged or smoothed over a number of "storms'.

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Advantages:

1. Quick and easy

2. Yields temps in many cases similar to that of a GCM (does one thing and does it well.)

Disadvantages

1. Only yields temperature - nothing else.

2. No real treatment of dynamics, only parameterized.

3. no real longitudinal resolution

4. appear to be quite sensitive compared to complex models (overly sensitive to feedbacks).

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The Radiative Convective model (RCM)

Compute vertical profile of an atmosphere column for generic point. Yields Temps, radiative fluxes and vertical distributions of mass.

Detailed computation of SW and LW radiation, and convection to remove near surface instability (since surface heats more readily than atmosphere) Must include convection to relieve wild lapse rates near surface due to surface heating.

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Advantage:

1. Most detailed computation that can be made at present for vertical distributions of atmospheric temperature Frequently uses radiation schemes as complex or more than that of GCMs.

2. Quick and easy to run

Disadvantages:

1. No dynamics, no horizontal advections.

2. No latitude or longitude resolutions.

3. Only a few climate variables calculated.

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SDM - a true dynamic climate model

A genuine climate model that solves directly for climatic means and variances on monthly to annual time scales.

Includes a computation of surface temperature similar to that of EBM with radiation code as complex as an RCM.

Includes explicit computation of dynamics, although synoptic scale events must be parameterized in terms of climatic means and variances.

Generally can give vertical and latitudinal quantities, but no longitudinal resolution.

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Advantages:

1. Simple and easy to run

2. Computes the most relevant climate variables, such as hydrologic cycle.

3. accounts for radiation, dynamics, and

sfc

. processes.

4. is true 'climate model'

Disadvantages

1. No longitudinal resolution.

2. Dynamic process and standing eddy parameterizations only, not well understood or parameterized.

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The General Circulation model

A PE model, solves for thermo and dynamics of the atmosphere for instantaneous or daily time scales.

Essentially same as forecast models, yields arbitrary 'weather" data that are post processes to yield model climate stats. (just like observations!)

relatively fine resolution in the vertical

(25 – 50 layers

), but relatively coarse in the horizontal 2 - 8 degrees in both latitude and longitude, but 1 x 1 simulations are

common, sometimes less.

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Explicitly computes radiation dynamics, and thermodynamics on hourly time scales (shorter should still be parameterized).

surface temperature computed as in EBM with actual advection replacing 'mean diffusion'); radiation as in RCM.

Advanages

:

1. most comprehensive model currently available (the plethora of physics makes it attractive to climate people).

2. full vertical latitudinal and longitudinal resolution (we still may not understand enough about standing eddies to know whether or not it's good).

3. most comprehensive between climate components and feedbacks.

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Disadvantages:

1. Very costly to run, or slow or complex.

2. Massive amount of output, makes analysis/interpretation difficult.

3. large number of parameterizations leads to considerable 'tuning' making predictive capabilities uncertain. OK for present

wx

, but for use on past or for future climates, use is uncertain.

4. Not a genuine climate model. Solves for weather, which is post-processed to get a climate.

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New Topic: Climate Change

This is certainly the topic of the day considering the controversy surrounding anthropogenic release of CO2, which in effect changes the composition of the atmosphere. Thus, this forcing can be considered "external". However, if humans are part of Earth-Atmosphere system, couldn't we argue internal?

If we have defined climate in terms of means, variations, and higher statistical moments, then climatic change can be defined as changes with respect to time of these means, variances, and other moments.

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Another, more precise way of looking at the issue: We have made the assertion that our definition of climatic averaging required that we average over a period with little variability, or that for any change in the statistical characteristics with time: which is typically 30 year averages. However, this is about one generation in lifespan. when we apply to a climatic averaging period of one month to several decades.

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It then follows that there are two possible ways for: 1) For time scales shorter than a climatic averaging period, e.g., hours to several days (synoptic period). This is the realm of weather and we won't consider. On this realm, we assume balance between two large forcings (PGF = CO).2) For time scales of longer than a climatic averaging period, e.g. hundreds to millions of years, this is the realm of climate change! Thus we assume: for time periods longer than a standard climatic means. Usually a balance between smaller forcings.

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Then, the next question we must then answer:

1) Does climate change actually occur?

Even a most cursory or rudimentary examination of the geologic record suggests this.

Example:

1) Medieval Warm (800 - 1350 Anno Domini)

2) Climatic Optimum 5 - 8 KYA

3) 2 MYA Pleistocene Ice Ages (at least 6, see articles in "Science") and

interglacials

.

3)

Cretacious

warmth (68 - 130 MYA)

4) The cool, glaciated

Permo

-Carboniferous (~300 MYA)

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KEY Issues in climate change:

1) Does climatic change operate equally over all time-scales, or are there time-scales that are preferred (e.g., have large climatic variability) and those that are not preferred.

a. One answer can be gleaned by looking at the climatic spectrum. Relatively little change takes place on time scales of a few hundred years. Considerable climate change on decadal scale (PDO, NAO, AO). Considerable change take place on time-scales of a few thousand years to 10 - 100 KY (

Milankovitch

). Relatively little on time scale of MY, with considerable change on 10+ MY. Thus, we see considerable preferential scales.

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2) Is climatic change global or regional?

This question is difficult to address. Popular perception fueled by global warming proponents is that it's global. Could be given

teleconnectivity

. Even some atmospheric scientists tend to think in global terms. If one looks at climate in more detail, one tends to find that the differences are primarily regional.

For example, the

pleistocene

Ice ages affected mostly mid and high latitude NH, with Equatorial region and SH unaffected, relatively. Global temperatures dropped about 2 C (see how sensitive climate can be? How close we are to ice age?), but this did not occur equally

everywhere

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Conversely, recent warmth,

1.0

C last 130 years, nothing in tropics, so far confined to polar regions in NH and SH (although parts of Greenland actually colder!). The

Cretacious

warmth occurred primarily at high latitudes with little impact on tropics.

Even climate change due to globally uniform increases in CO2 (such as currently occurring?) has a large regional differences, e.g., high latitudes respond more than low latitudes. Thus, it is reasonable to assume that high latitudes are more sensitive than tropics.

Regionality

It has been shown that, where ice sheets dominated delta T was on order of 30 - 50 C cooler, but more of the globe had smaller changes.

High latitudes should be more sensitive, since Tropics are heat surplus regions and poles deficit. Thus, how efficiently mixing and transport is done by atmosphere and oceans, with dictate climate.

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3) Is climatic change primarily forced by external causative factors (forced modes, or changes in BC's), or Internal dynamics (non-linear interactions, feedbacks) of the climate system (free modes)?

It appears both are very important. Obvious external

forcings

:

Milankovitch

orbital parameter changes, Continental Drift, changes is Solar parameter, Volcanism, etc. However, it also appears that interactions between climatically important variables (especially those with fairly long-timescales) may also lead to climatic change. This could represent changing atmospheric composition (CO2?) or changing Deep Ocean Temps.

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Can we distinguish 'fast response' climatic variables from 'slow response' variables?

Yes, and this is a crucial distinction. Fast response climate variables can be defined as those with equilibrium times shorter than 10 - 30 year averaging period. This includes more common variables such as, atmospheric temperatures, SSTs, precipitation, etc. Slow response variables on the other hand have characteristics equilibrium times longer than a standard averaging period. These include, glacial ice, deep ocean temp., and atmospheric CO2.

Thus, variables which act on time scales longer than your averaging period, no matter what the space scale are climate change variables!!!

So if we built a climate model based on "fast" response variables (e.g. SDM) and assuming balance on scale of climate, then theoretical models of climate change must be built in terms of the slow variables.

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How do our balance compensate for climatic change? What do we do if: Recall:

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And where; And;

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Then as before: Total EnergyNow integrate this globally: area of earth

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when we perform these operations we get the beastly equation: net radiation in water mass phase changes geothermal surface topography where:

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Let's make a few reasonable assumptions to aid in our scale analysis:Ma = constMl = ConstMw + Mi = Const or that:

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Now divide through by the area of earth to change from W (Watts or total global power) to W/m2 (globally averaged value) to get another beastly equation:

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Now we will do our scale analysis on the individual terms in the equation. Use the following parameters:

 

Specific Heats; Latent heats

 

c

p

= 1 x 10

3

J kg

-1

K

-1

L

f

= 3.34 x 10

5

J kg

-1

c

w

= 4.2 x 10

3

J kg

-1

K

-1

L

v

= 2.5 x 10

6

J kg

-1

c

i

= 2.1 x 10

3

J kg

-1

K

-1

c

l

= 1.6 x 10

3

J kg

-1

K

-1

 

 

Surface Areas densities

s

= 5 x 10

14

m

2

r

a

= 1 kg m

-3

s

w

= 5 x 10

14

m

2

r

w

= 1000 kg m

-3

s

l

= 5 x 10

14

m

2

r

i

= 920 kg m

-3

s

i

= 5 x 10

14

m

2

(at T = 275 K)

 

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Mass of each where seasonal cycle is important:Ma = 5 x 1018 kg Mw = 1.4 x 1021 kg Ml = 1.6 x 1018 kg Mi = 26 x 1018 kg (now) Mi = 77 x 1018 kg (18 KYA) We can assert that several terms in the equation will be relatively very small and hence negligible:  ignore change in cloud ice over time since the reservoir is small

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 topography changes are small over 10 KY scale except for ice sheets  KE changes are small except in the atmosphere but these changes are small over time.Last time we derived a relationship that provides information about the processes that causes long-term climate change. We started talking about what kind of assumptions we wanted to make. Let's show example of use with ice age theory (Energy equation).

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An important aside: (this will have very grave implications for modeling. This is at the heat of climate change!)That is, How to evaluate:this is the change in NET GLOBAL ENERGY due to the presence / absence of large ice sheets. In otherwords, the growth or decay of ice sheets with time will lead to a net rate of change in the net latent heat of fusion in the climate system. When ice forms  heat is liberated for other parts of the system. When ice melts energy is drawn in.

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Thus how do we evaluate: For our ICE Age Pleistocene situation, a reasonable guess is that 50 x 1018 kg of ice accumulated / melted in about 10,000 years.Then:Joules of energy! (sun outputs 3.9 x 10^26 J/s).

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Change over 10,000 years: divide former # by 3 x 10

11

s = 5 x 10

13

Watts

 

This is the total amount of energy per unit time that must be accounted for in a physically-consistent theory of the ice ages (e.g., one that adheres to conservation of energy). However, this comprehensive theory of climate or climate change must also be global in nature (since all components of the climate system are open with respect to transfers of energy and mass).

 

This is why, for example, we integrated our energy balance equation over the globe (and divided by sigma to express in terms of per unit area).

Then (divide by surface area of earth):

 

5 x 10

13

Watts

____________

 

5.1 x 10

14

m

2

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= 0.1 W/m2 !!!! 0.1 W/m2 is a very small number in terms of net energy gain/loss per unit area per unit time, but one that represents THE Minimum LEVEL of sensitivity to which any quantitative, theoretical model of the ice age climatic change MUST adhere.

Even assuming near-perfect parameterizations, the best current GCM's can do is about +/- 1 W/m2. Thus, a truly deductive model of the ice ages that is globally consistent is not at present possible.

The same applies to Global warming theories!

Back to scale analysis:

We need to assign typical 'ice-age' temperature differences:

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= 10 C (keep large to constrain conservation) This term becomes important, but the real change globally was on order of 5 c or less. = 10 C = 10 C = ? determine from scale analysis, We'll keep this ???? for now since ocean is 3 orders of magnitude bigger than other components of climate system.

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Also, G, the geothermal flux, has a reasonable global value of about 0.06 W/m2 (almost that of 0.1 W/m2). No reason to assume it's changed much in 18 KY (unidirectional flux). Also, not negligible for long-term climate change!

We'll choose to ignore KE, but what about PE

(

f

)?

our main source of changing

f

will be elevation of the ice sheets themselves.

ow

, let's re-write our scaled equation, listing characteristics values for each term:

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Now, let's re-write our scaled equation, listing characteristics values for each term:10-3 10-4 10-3 ????atms T change land T change ice T change ocean T change10-3 ????? 10-1dry static energy net radiative flux geothermal

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The middle terms, Latent heats for water and ice and land are: 10-1, 10-1, and 10-5.

So, ice ages would seem to be caused by ocean temps, net radiation or both (BC's). They are same order of magnitude as 0.1 W/m2.

The 2 largest terms we can evaluate (Geothermal and latent heating) are both on order of 0.1 W/m2. Thus we can discount all terms with scales of 10-3 W/m2 or less as being unimportant for long-term balance.

Also, G is unidirectional (positive flux from earth to

sfc

.). Latent heat of fusion can be = or - .

then either of these terms must be on order of 0.1 W/m2.

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In other words, if we balance Lf with some combination of: If we assume N = 0 or always in radiative balance (always in even small imbalances), then the world oceans (including deep waters) must have been 3 - 4 C warmer during ice ages than present (seems counter - intuitive, but could easily happen since seas ice extent can shut down deep water production (shuts down convective mixing).

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However, there is no reason to assume perfect radiative balance. We cannot at present distinguish further between the relative importance of N and: Think about this for test. Which terms if any might be important in "global warming"?

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Modeling and Long term climate change:

 

Inductive versus Deductive Reasoning

Inductive

when we say X is important, Y is important, and Z,W, and V are small, thus

 

X =

Y

 

This

is scale analysis and we’ve just finished this. Our

Pliestocene

Ice age works well for this.

A priori

reasoning

 

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Deductive reasoning

takes a model result and figure out or calculate which processes are most important. A posteriori reasoning.

A GCM can be used to do this or the Omega equation can be used to get down to Q-G form.

Given model tolerances; deductive sucks.

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The problem can be posed as one of two questions:

1) How can we quantitatively computer the location and volume of

icesheets

at 18 KYA (during last glacial maximum?)

Note the question is stated in the equilibrium sense, that is, can we determine the MEAN CLIMATE STATE for a point in time in the past. While such ability may be an important point in the understanding of past climate, e.g.

paleoclimate

, it is not really a question of climate change per se.

In other words, we can solve the above stated problem for 18 KYA with:

that is w/variables of 18 KYA at their values of the time and then not changing!!! So we're not investigating climate change since we're not commenting on how climate got from A to B!

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2) How can we quantitatively compute the change in ice sheet location and volume as the climatic system transitions between a state of little ice (e.g. the present) to a state of maximal ice (e.g. 18 KYA), and back again. Now we have stated question in a dynamic sense, e.g., we have stated it such that:

Now it's a question of climate change!!!!

What is the most fundamental task we need to accomplish in constructing a model that can be used to provide a physically consistent answer to question 2?

For ice age, the greatest importance is that of net annual snow accumulation - a positive net mass balance (accumulation > melting annually) somewhere on the globe is required for there to be a glacier / ice sheet in the first place, while changing spatial patterns of net snow accumulation must be indicative of changing ice sheet size and location.

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Calculation the net annual snow mass balance requires computing;

1) snowfall, which ultimately requires an accurate computation of global hydrologic cycle, as well as precipitation physics

2) Snowmelt, which requires an accurate computation of the local surface energy balance. It's probably more important to get this right since it's a

contiuous

process!!!

Questions 1 and 2 involve fast response variables at face value (

Ts

,

Precip

, Q, V, P, etc.), and these have equilibration times which are 'instantaneous' relative to climate averaging.

Regardless, the means and variances change w/ time. The question is why? Which slow processes cause this?

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ATMS 8600 Dynamic Climatology

So we must resort to larger-scale processes (slow response variables) and/or external forcing to account for the necessary changes that must have taken place in the fast response climate variables. It's possible only external forcing (e.g.,

Milankovitch

forcing) may be involved.

one slow

respose

climatic variable of obvious importance in the ice sheet movement itself (they do not grow or shrink

insitu

.

You should not think of these as static and growing or shrinking only due to local changes in net snow accumulation. Ice can be thought of as a very viscous (like glass! lot's of internal friction) fluid that flows in response to internally generated pressures.

Ice flows from ice source (net mass gain, positive balance) to ice sink (net mass loss, or negative balance) via horizontal transport. This is how ice sheets can exist in areas where it might not otherwise be possible (Canada huge source region for US glaciers?)

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Thus in an equilibrium sense, one can balance sources and sinks vs. transports of ice. but, it's the changes in these quantities w/r/t time that we are interested in.

Also, it should be obvious that the flow rate of glacier will depend on net mass buildup and decay on fringes, the greater the difference the faster the ice flow rate.

Also, an ice sheet of 1000m or more in height will act effectively like a mountain (Appalachians and Ozarks?) and modify the local, region, and even hemispheric weather patterns accordingly affecting snow balances. North American ice sheets act like Tibetan plateau. (Increasing topography and blocking)

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Thus we mean to say that: also, we can speculate that variations in slow response variables are also important in explaining ice ages, namely atmospheric CO2 and deep ocean Temperatures. Thus, we can write our ice balance equation as; Internal variables to climate! ice balances Atmospheric deep ocean fluxes CO2 temperatures

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However, if this is the case, we must also account for variations in CO2 and T, which also depend on one another (non -linearly). So we can write the following closed set of equations for the dynamical system (Similar to Lorenz and the Equations of Motion (Atms 9300):

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ATMS 8600 Dynamic Climatology

note we've just set the table, we have not eaten yet, thus we have not specified the functional forms for f1 – f9 (or even what the net sign is!!!) We must assume they all depend on one another until proven otherwise.

Before going further, we must raise the issue of external forcing. This is particularly appropriate for the Pleistocene Ice Ages, since the 3 main periodicities seen in the isotope record of glaciation, 19 - 23 KY (procession), 41 KY (obliquity), 100 KY (eccentricity), correspond to the three periodicities of

Milankovitch

earth orbital forcing.

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If you schematically denote this forcing as 'R', we also not that ice ages cannot be considered as merely a forced linear response to insolation changes because primary ice age 'power' is 100 KY while

Milankovitch

'power' at is extremely small.)

Although

not as obvious as

Milankovitch

forcing (R), other external agents may be important, like volcanism, and the geothermal flux, will denote all the as 'F'.

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ATMS 8600 Dynamic Climatology

Equations:

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Now there are other issues that need addressing:1) How do we obtain functional forms for f1 – f9? For the most comprehensive approach we would like to deduce these functions from first principles (1st law and NS equations).presumably, these should include rates of change such as:

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ATMS 8600 Dynamic Climatology

that express the changes in slow response variables and their impact on one another, and on rate constants such as; which express the effects of slow response variables on fast response (non-linear terms). Thus we must include the fast response variables at least implicitly since they are essential to snow mass balance. We already know though that the current state of modelling does not allow us to trully construct a deductive model of climate changes, since it requires sensitivity to 0.1 W/m2.

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ATMS 8600 Dynamic Climatology

Thus we can do only one of two things

1) resort to the geologic record and attempt to infer from it relationships between slow variables (inductive),

or

2) perform 'what if' sensitivity studies to help evaluate the reasonableness of postulated functional combinations.

A better posed question would concern the effects of slow response variables on fast response variables, since these can be estimated in a deductive fashion using equilibrium models of climate, e.g., the SDM or GCM

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ATMS 8600 Dynamic Climatology

we can calculate OK right now, we’re screwed right now!Second is the question of weather orbital insolation or any other external forcing is a necessary or sufficient condition for the ice ages, or whether they are the result of internal behavior of the the climate system, perhaps 'paced' by insolation changes. The last question is tatamount to asking if the cyclical nature of the ice ages would occur without Milankovitch forcing? This question has fierce proponents for both Yes and NO. In reality I think a combination of the two are important, or equally important, but it will be difficult given today's knowledge to determine to what extent they are important relative to each other.

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Models of Long Term Climate change:

1. Self oscillating (with and/or without external forcing). (as opposed to a non-oscillating)

a. Saltzman - Moritz (not a true self-oscillator but a first step along the way) (it's unforced, good first step).

See

Tellus

, 32, 93 - 118 (1980)

Develop similar to Lorenz:

Fast response variables:

atms

and SST,

atms

. heat fluxes

Slow response: sea ice extent, and deep ocean temp (2 types: 1. self oscillating or non-linear. 2. Linear)

No external forcing!!

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Positive Feedbacks: (

i

) ice albedo feedback posed as a sensible heat flux 'rectifier' feedback involving synoptic air mass heat exchange as a consequence of atmospheric

baroclinicity

. (the more the sea ice, the greater the

baroclinicity

, the greater the heat loss from the ocean to the atmosphere tending to cool the ocean, and leading to more sea ice).

 

(ii) long wave emissivity changes due to CO

2

changes postulated to arise in response to variations in mean ocean temperatures. Cooler ocean hold more CO

2

, warmer hold less, thus, it's a positive feedback

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Negative feedbacks: (

i

) insulating effect of seas ice on deep ocean temps. leading to warming and having less seas-ice.

(ii) change in amount and path length of solar radiation at ice edge. (As sea ice moves

equatorward

more heating/melting can take place due to SW radiation).

Model expressed mathematically as a coupled, autonomous polynomial system of 6th degree governing seas ice extent and deep ocean temp. this is cumbersome and abstract, so won't show.).

Results: For certain reasonable parameters, a damped oscillation with a period of 1 KY occurred. + and - feedbacks too strong, and - damped too much. Self-oscillating, doesn't grow or decay. For other values unstable behavior. Instabilities occurred for large departures from equilibrium. Model was a failure, but moving in the right direction.

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Saltzman and

Sutera

model:

Improvement on previous model. JAS, 41, 1984 736 - 745. JAS 1987, 44, 236 - 241.

Model is based on the assumption that key slow response variables are continental ice mass, seas ice, deep ocean temp., and carbon dioxide.

They then assumed total ice = seas ice + land ice. They also added CO2.

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ATMS 8600 Dynamic Climatology

Here's the dynamical system:

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ATMS 8600 Dynamic Climatology

This is an unforced system. they assumed further that: a2 = a3 = bo = b2 = b3 = c1 = 0essentially performing an ad-hoc sensitivity analysis.Note that I, u and Tw are assumed to be departures from some equilibrium.Final forms:

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ATMS 8600 Dynamic Climatology

Now there are 7 coefficients. that must be prescribed empirically.

The model does indeed demonstrate (with appropriate coefficients) the 100 KY glacial-like cycle, with explicit phase relations between each variable. The model also provides for a transition at about 1 MY between low-amplitude and high amplitude ice-age oscillations (like synoptic-scale

vascillation

?), which are also suggested in the ice age record.

Saltzman, Hansen, and

Maasch

, 41, JAS, 3380 - 3389

Saltzman,

JClim

, 1, 77 - 85.

Same basic system, but they replace CO2 with marine ice.

They get good phase lock to ice fluctuations inferred from O-18 deep sea records.

Note

Milankovitch

(external) forcing is not necessary or sufficient to produce ice age cycles, but as studies do show it IS necessary to get the proper phasing of Actual

plietocene

iceages

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Internal forcing gets cycles - Milankovitch locks it in.2. Forced, non linear models  orbital forcing is necessary and suffient to force ice age cycleSingle variable: Imbrie and Imbrie, is elegant but wrong, see Science, 207, 1980, 943 - 953.volume of ice equation: F = ext. forcing

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F = ext. forcingI and F, assume linear and non-dimensional form: where T is characteristic time constant.They get reasonable values for T of 3 to 30 KY, but they noticed that glacial decay operated much more quickly than glacial buildup (draw-down rapidly, build slowly- the familiar saw-toothed (non-linear) pattern we observe in geologic record.

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ATMS 8600 Dynamic Climatology

They get reasonable values for T of 3 to 30 KY, but they noticed that glacial decay operated much more quickly than glacial buildup (draw-down rapidly, build slowly- the familiar saw-toothed (non-linear) pattern we observe in geologic record.So they rewrite: if F >= I decay if F <= I build-up

Slide411

ATMS 8600 Dynamic Climatology

Tm is now mean time constant of the system and b is a non-linear coefficient (0 < b < 1) defined in such a way that large values of b correspond to larger time differences between glacial onset and decay. As b varies between 0.33 and 0.66, the ratio of the time constant varies between 2 and 5 (glacial decay is happening 2 to 5x as fast as buildup). This is the essential non-linearity. Tm and b are both tunable parameters.

Specify reasonable values lead to a model that predicted ice volume (in absolute and power spectrum) in reasonable form for the last 400 KY. This model is useless beyond that.

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In closing, future modeling of climatic change will likely process in the following

1)

Continued construction

of super-climate ,models (super GCMS or SDMS) that take into account fast and slow response climatic variables and hence can be integrated through geologic time. For many, this is the ultimate goal, but not clear if it's attainable. 1) much computer time, 2) recall sensitivity needed!! (Typically a combination of fast and slow response variables lead to a "stiff" system.

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2) Construction of more sophisticated but simple (forced or unforced) dynamical models aimed at a better understanding of fundamental physical processes and climatic feedbacks. Side ways progress.

3) Combination of 1) and 2). It's not yet clear we can do it. also, impossible to say which is better! These are all different approaches.

The

End

!!!

Конец

!!!!!!

!

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