Atmospheric Science 8600 Advanced Dynamic Climatology

Atmospheric Science 8600 Advanced Dynamic ClimatologyAnthony R. Lupo302 E ABNR BuildingDepartment 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

ATMS 8600 Dynamic ClimatologySyllabus **

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.

ATMS 8600 Dynamic Climatology5.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!

ATMS 8600 Dynamic ClimatologyThis 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.

ATMS 8600 Dynamic ClimatologyRecall in meteorology, we tend to divide phenomena by scale based on what processes are important to driving them.

Table 1

ScaleTimeSpaceForceWeatherPlanetary7 14 days6000+ kmCo, PGFJet streamSynoptic 1 7 days2000 6000 kmCo, PGF, Fric, BouyLow pressureMeso1 - 24 h 10 2000 kmBouy, PGF, FricFrontsMicro< 1 hr< 10 kmBouy, Fric, PGFThunderstormATMS 8600 Dynamic ClimatologyGeneral 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.

ATMS 8600 Dynamic ClimatologyClimatology 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).

ATMS 8600 Dynamic ClimatologyKey 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

ATMS 8600 Dynamic ClimatologyWe'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.

ATMS 8600 Dynamic ClimatologyThe 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)

ATMS 8600 Dynamic ClimatologyThe earth-atmosphere system, courtesy of Dr. Richard Rood.(http://aoss.engin.umich.edu/class/aoss605/lectures/)

ATMS 8600 Dynamic ClimatologyAside:

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.

ATMS 8600 Dynamic ClimatologyAnother view of the climate system

ATMS 8600 Dynamic ClimatologyAnother view

ATMS 8600 Dynamic ClimatologyEach 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. ATMS 8600 Dynamic ClimatologyOk, 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.

ATMS 8600 Dynamic ClimatologyRecall, 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') > 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.

ATMS 8600 Dynamic ClimatologyFor climate:

fast response (diagnostic) atmosphere

slow response (prognostic) ice sheets, ocean

(for example, if ice sheets grow, we know how atmosphere will behave)

ATMS 8600 Dynamic ClimatologyDiagnostic or prognostic?

ATMS 8600 Dynamic ClimatologyClimate 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".

ATMS 8600 Dynamic ClimatologyThe scientific Method:

1.Collect Data 2.Investigate the Issue3.Identify the Problem4.Form Hypothesis5.Test Hypothesis6.Accept or Reject hypothesis based on conclusions7.If reject, goto 28.If accept, move on to the next problem.

ATMS 8600 Dynamic ClimatologyTwo 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)

ATMS 8600 Dynamic ClimatologyOcean - 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!

ATMS 8600 Dynamic ClimatologyFeedback 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) ATMS 8600 Dynamic ClimatologyExamples: Double CO2

-Positive feedback-

Linearly increase CO2 to double the amount has a very small effect on climate temp. response or forcing.

It's the other feedbacks that amplify "global warming"

CO2 is a "potent" greenhouse gas, but a trace gas.

ATMS 8600 Dynamic ClimatologyWater Vapor is more plentiful and 30 times more potent "greenhouse gas" (molecule per molecule, due to vibrational, rotational absorption bands).

Increased CO2, (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.

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.

ATMS 8600 Dynamic ClimatologyTemperature 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

ATMS 8600 Dynamic ClimatologyHow 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!

ATMS 8600 Dynamic Climatology3 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.

ATMS 8600 Dynamic ClimatologyTypically 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.

ATMS 8600 Dynamic ClimatologyAlso 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.

ATMS 8600 Dynamic ClimatologyThe 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.

ATMS 8600 Dynamic ClimatologyRecall 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:

Notationc = carrier fluid a = atms. w = ocean i = ice

j = trace constituents

ATMS 8600 Dynamic ClimatologyXc = mass concentration of carrier fluid

Xj = mass conc. of jth trace constituent

Total density of an arbitrary volume in the climate system

Often times, Xc >> Xj

ATMS 8600 Dynamic ClimatologyAlso, to be pure, we need to say for example = Xc / Volume

mass fraction or mixing ratio (trace constituents):

Remember in Atmospheric Science? Mixing ratio: Mv / Md?

ATMS 8600 Dynamic ClimatologyContinuity equation (general) carrier

see where atmospheres. 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 doesnt go to 0!

ATMS 8600 Dynamic ClimatologyXc (atmospheric density - Continuity)

(From Atmospheric dynamics)

Water Vapor (trace gas):

Sc Source + sink

Xc q (specific humidity Mv / (Md + Mv)

ATMS 8600 Dynamic Climatologytrace continuity equation

Vc = velocity of carrierVj = velocity of trace

Sc = sources sinks of carrierSj = sources sinks of trace

ATMS 8600 Dynamic ClimatologyRecall from vector calculus we can put into advective forms

As in atmospheric version

ATMS 8600 Dynamic ClimatologyAlso, we can decompose Vj as follows

where,

mj = molecular diffusion

wjk - fall due to potential gravity field

then,

ATMS 8600 Dynamic ClimatologyWhere,

ABC

A = turbulent eddy fluxB = mean molecular diffusion fluxC = mean sedimentation

ATMS 8600 Dynamic ClimatologyAgain, water vapor:

j = H2Ov

Xj = S = Evap + Precip

One 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.

ATMS 8600 Dynamic ClimatologyFinally one could consider the total fluid volume (carrier + trace substances combined!).

ATMS 8600 Dynamic ClimatologyThe 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

ATMS 8600 Dynamic ClimatologyIn component form (spherical coords)

ATMS 8600 Dynamic ClimatologyThermodynamic 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

ATMS 8600 Dynamic Climatologydh (or dq) = diabatic heating, rate of addition of heat

du = rate of change of internal energy per unit mass

dw = rate of work on unit mass by compression (pressure work term)

In the atmosphere:

NOT SAME AS

ATMS 8600 Dynamic ClimatologyIn these equations:

= 1 /

OK, now we can rewrite the first law:

ATMS 8600 Dynamic ClimatologyTake 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:

ATMS 8600 Dynamic ClimatologyWe 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).

ATMS 8600 Dynamic ClimatologyThen, 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.

Componentform of du

Dry air

H2Ov

ATMS 8600 Dynamic ClimatologyComponentform of du

Dry air + moist.

Liquid water:

Ice

Lithosphere

ATMS 8600 Dynamic ClimatologyEquations 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)

ATMS 8600 Dynamic ClimatologyThis 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: Earths Oceans T, , P, and Salinity (S)

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):

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

ATMS 8600 Dynamic Climatologyterm 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!!!

ATMS 8600 Dynamic ClimatologyEquation 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!!

ATMS 8600 Dynamic ClimatologySo Review fundamental equations of Geophysical Fluid dynamics:

A. Continuity:

B. Equation of Motion

ATMS 8600 Dynamic ClimatologyC. Thermodynamics:

D.Equation of State (Gasses)

ATMS 8600 Dynamic ClimatologyRecall concept of Reynolds Averaging

Where X is much smaller than Xbar

Instantaneous wind: (u)

(where prime is a departure from time average, say yearly, seasonally, or any time scale appropriate to the phenomena youre studying.)

This averaging can be carried out no matter what scales you examine!

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!

ATMS 8600 Dynamic ClimatologySo, the product of U and V:

So, if we average the product, then? What must happen to middle terms?

ATMS 8600 Dynamic ClimatologyWe can, then naturally apply the same procedure to space averaged data to get:

[uv] = [u] [v] + [u* v*] (2)

how about?

ATMS 8600 Dynamic ClimatologyOK, 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!)

ATMS 8600 Dynamic ClimatologyWhat does (3) mean?

A B C

LHS = 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.

ATMS 8600 Dynamic ClimatologyWe 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)

ATMS 8600 Dynamic Climatologyconsider 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

ATMS 8600 Dynamic Climatologyw 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;

ATMS 8600 Dynamic Climatologyassume atmosphere is well behaved, reverse order of integration and differentiation (and assume wt = 0):

AB C

we call dry mass integral (also a,b,c are boundary recall)

ATMS 8600 Dynamic ClimatologyWhere 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

ATMS 8600 Dynamic ClimatologyFirst decompose:

Then zonally average:

Then time average (bar)

ATMS 8600 Dynamic Climatologylets 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!);

ATMS 8600 Dynamic ClimatologyGlobal 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 well look at each sub component of the system and their interactions, since water mass is everywhere.

ATMS 8600 Dynamic ClimatologyStart with continuity:

well say Xj = Xv (atms. vapor)=Xn (cloud water)=Xw (surface water land and oceans)=Xi (surface ice)

ATMS 8600 Dynamic ClimatologyThen for vapor (1),

Evap Subli.lvsvfor cloud water (2)

Cond. Ablat.

ATMS 8600 Dynamic ClimatologySurface 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!

ATMS 8600 Dynamic ClimatologyAs before we can integrate each in the column:

use the following notation:

example condensation!

ATMS 8600 Dynamic ClimatologyThen for water vapor equation:

horiz and vert cloud surf vapor flx formation

Redefine Xv (mass of water vapor mv):

ATMS 8600 Dynamic ClimatologyThen 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)

ATMS 8600 Dynamic ClimatologyDiagram

ATMS 8600 Dynamic ClimatologyThen (Atmospheric water vapor):

precipitablehorizontal Evap precip Cloud watertransport water

Using the same procedures as before, we change variables and handwave.

ATMS 8600 Dynamic ClimatologyCloud water:

transport ofprecip phaseclouds changesSurface Water (Soil science folks like this):

Evap Meltingliquid precip

ATMS 8600 Dynamic ClimatologyDistinguish between land and ocean

In 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!

ATMS 8600 Dynamic ClimatologySurface Ice (fundametnal cont. equation of glaciology)

snowsnowsnowevapmelt

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!

ATMS 8600 Dynamic ClimatologyGlobally integrate the four and all divergences vanish (transports disappear of course!).

ATMS 8600 Dynamic ClimatologyThen 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

ATMS 8600 Dynamic ClimatologyFor the Atmosphere we can show the General Circulation version of the water balance equation versus climate (whats typically used).

Dynamic Climates

General Circulation

ATMS 8600 Dynamic ClimatologyWe covered this in General Circulation recall primarily done in tropics by mean motions and in the mid-latitudes by eddy motions.

ATMS 8600 Dynamic ClimatologyAngular momentum Budgets

Recall 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:

ATMS 8600 Dynamic Climatologyr 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:

ATMS 8600 Dynamic Climatologyglobally integrate:

Where:

= mean angular velocity of the Earth

I = Moment of Inertia

ATMS 8600 Dynamic ClimatologySeparate into Earth and atmosphere parts.

then:

so:

(1)(2)(3)(4)

ATMS 8600 Dynamic ClimatologyLike the water budget, well retain terms neglected in the General Circulation, that is (1) and (2), and (3).

We'll eventually assume that the external torques are negligible 0

What are each of the terms?

ATMS 8600 Dynamic ClimatologyTerm (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!

ATMS 8600 Dynamic ClimatologyChange in length of day ugh, those long days!

Journal of Geophysical Research: The two types of El-Nio 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 Nio makes the day longer by about 1 x 10-6 s

Also see: http://www.aoml.noaa.gov/general/enso_faq/

ATMS 8600 Dynamic ClimatologyTerm (4) This is the momentum budget (13 terms) from Atmospheric Science 8400

A 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

ATMS 8600 Dynamic ClimatologyTerms (1) and (3) together reflect the mass shifts of the Earths 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)-

ATMS 8600 Dynamic ClimatologyRecall, 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).

ATMS 8600 Dynamic ClimatologyRecall Atms 8400 equation:

ATMS 8600 Dynamic ClimatologyRecall 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).

ATMS 8600 Dynamic ClimatologyThus, 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)

ATMS 8600 Dynamic ClimatologyRecall 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.

ATMS 8600 Dynamic ClimatologyOver 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 Earths radius is constant)

.

ATMS 8600 Dynamic ClimatologyThen, 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 requirementsATMS 8600 Dynamic ClimatologyTransport 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.

ATMS 8600 Dynamic ClimatologyYou 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.

ATMS 8600 Dynamic ClimatologyHeat 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 1st 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.

ATMS 8600 Dynamic ClimatologyFundamental 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.

ATMS 8600 Dynamic Climatologyterm (1) is the pressure work term

term (2) dissipation (friction)

term (3) radioactivity (Uranium, etc.), chemical, geothermal

term (4) diabatic heating

ATMS 8600 Dynamic ClimatologyDiabatic 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)

ATMS 8600 Dynamic ClimatologyInternal energy per unit mass:

and

Expressions for e:

Component Internal EnergyPhase Changes

Atms (a) vapor freezing

ATMS 8600 Dynamic ClimatologyComponent Internal Energy Phase Changes

Ocean (w,i)

Sea ice

Land (l)

Freezing of ground and/or lakes

ATMS 8600 Dynamic ClimatologyExpression 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).

ATMS 8600 Dynamic ClimatologyWe can decompose:

vaporice

These are Fluxes of Latent heat of vaporization and ice which can be decomposed into:

ATMS 8600 Dynamic ClimatologyWe 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:

ATMS 8600 Dynamic ClimatologyThen add to 1st law of thermodynamics:

To get Bernoullis equation: (Recall from Atmospheric Dynamics)

ATMS 8600 Dynamic ClimatologyNow we have an equation for dry static energy (Bernoulli's equation): Where we can set the energy terms as: (zh Ive run out of Greek letters!)

= e + KE + gz

Let's get ready to integrate, and put equation in flux form!

ATMS 8600 Dynamic ClimatologyHere 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 water

b - region below surface where seasonal changes are important

c - 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.

ATMS 8600 Dynamic ClimatologyRemember This?

ATMS 8600 Dynamic ClimatologyOn 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

ATMS 8600 Dynamic ClimatologyBottom boundary (Atmosphere):

Integrate:

ATMS 8600 Dynamic ClimatologyNow 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.

ATMS 8600 Dynamic ClimatologyLets re-define total energy:

and the horizontal fluxes:

synoptic-scale subsynoptic fluxesfluxes

what do these mean in the oceans? Cryosphere?

ATMS 8600 Dynamic ClimatologyAlso, 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):

ATMS 8600 Dynamic ClimatologyAnd:

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 Snowmelt

In the above expression:

H1 = Shortwave in (solar, space)

H2 = Shortwave out

H3 = Sensible heating (conduction and convection)

ATMS 8600 Dynamic ClimatologyAlso, 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):

ATMS 8600 Dynamic ClimatologyFor z1 = Zs, note from our earlier analysis of water vapor budget that the first term is Evaporation in LHR component, and second term is Snowmelt

In the above expression:

H1 = Shortwave in

H2 = Shortwave out

H3 = Sensible heating (conduction and convection)

ATMS 8600 Dynamic ClimatologyThen 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.

ATMS 8600 Dynamic Climatologya)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:

ATMS 8600 Dynamic Climatologyand 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.)

ATMS 8600 Dynamic Climatologyc)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)

ATMS 8600 Dynamic ClimatologyOK let's go ahead for atms:

Continue:

N approaches 0 on the scale of climate, but not so for daily values or for 1000s of years!

ATMS 8600 Dynamic Climatologynet radiation from the earth system

Remember! Sun does not heat the atmosphere primarily, it heats the ground which in turn heats the atmosphere!!!

So. Lets see the equation..

ATMS 8600 Dynamic ClimatologyAnd recall:

KE fairly smallPE + IE large (TPE from mixing ratio mixing ratio Gen circ)

Maybe not for Individual systems?

ATMS 8600 Dynamic ClimatologyWe 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

ATMS 8600 Dynamic ClimatologyPE + IE = TPE = rcpT

Now put back into our equation for energy:

Total Energy In the Atmospheric Column!!!!

ATMS 8600 Dynamic ClimatologyHorizontal transports (as shown in Atms 8400 Gen Circ):

Horizontal fluxes of: Dry static energyLatent Heats

ATMS 8600 Dynamic ClimatologyOn to Layer B!!!

Where the seasonal cycle is important:

ATMS 8600 Dynamic ClimatologyHeat equation in seasonal layer:

And:

latent heat of iceWhat happened for Latent Heat of Vapor?

ATMS 8600 Dynamic ClimatologyAlso; 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.

ATMS 8600 Dynamic ClimatologySo the above is:

then:

The above should be approximately constant, this the expression can come out of the integral

ATMS 8600 Dynamic ClimatologyAlso from continuity:

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

159ATMS 8600 Dynamic ClimatologyWhere:

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.

ATMS 8600 Dynamic ClimatologyYou can substitute for Pi using:

Lfqi

and then substitute continuity for a glacier (fundamental equation of glaciology):

ATMS 8600 Dynamic ClimatologyThis 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!

ATMS 8600 Dynamic ClimatologyThen finally we get:

Which, in the long term = 0!!!! (our fundamental requirement for balance!!!).

ATMS 8600 Dynamic ClimatologyThe 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

ATMS 8600 Dynamic ClimatologyThen 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.

ATMS 8600 Dynamic ClimatologyThis 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 Neils and Bos classes.

ATMS 8600 Dynamic ClimatologyWhen 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.) GCMs 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!!!

ATMS 8600 Dynamic ClimatologyLayer 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!!!

ATMS 8600 Dynamic ClimatologyThe equation:

ATMS 8600 Dynamic ClimatologyWe can group equations:

ATMS:

Subsurf:

Sub-surface all change slow with time!

ATMS 8600 Dynamic ClimatologyWe know the atmosphere changes quickly, that its 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.

ATMS 8600 Dynamic ClimatologyNow 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.

ATMS 8600 Dynamic ClimatologyRadiation and energy balance termsReview: 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.

ATMS 8600 Dynamic ClimatologyHs1 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.

ATMS 8600 Dynamic ClimatologyShortwave

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

ATMS 8600 Dynamic Climatologyrt - reflectivity above clouds

rs - surface reflectivity (albedo!)

X Opacity / absorptivity of atmosphere (up and down)

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

ATMS 8600 Dynamic ClimatologySchematic (clear sky):

ATMS 8600 Dynamic ClimatologyCloudy Sky Diagram:

ATMS 8600 Dynamic ClimatologyIf clear skies, then:

is the CO-Albedo!

ATMS 8600 Dynamic ClimatologyIf cloudy skies, then:

Now consider some of the parameters involved in Ht and Hs and what the typical values might be.

ATMS 8600 Dynamic ClimatologyFor 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 reflectivityr = 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.

ATMS 8600 Dynamic Climatologyrt = 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 = XCO2 + XO2 + XO3 + Xaerosols + Xvapor

ATMS 8600 Dynamic ClimatologyFirst three terms - fairly uniform spatially except for Antarctica in the springspatially 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

ATMS 8600 Dynamic ClimatologyIn 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 equatorP(f) path length as a function of latitude

P(f) = R(0) at equinox / R(f) at equinox or normalization factor.

ATMS 8600 Dynamic ClimatologyWhere 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:

ATMS 8600 Dynamic ClimatologyThen, based on empirical observations we make calculations of R.Xd = 0.06 (dry) Xv = 0.11 (vapor) X = 0.17 (total) Xn = 0.04 (clouds)

r = 0.14 (avg. earth albedo) rt = 0.05 (above clouds) rn = 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!

ATMS 8600 Dynamic ClimatologySo,

Short wave at sfc.:

Part 1: clear sky part of the diagram

Part 2:cloudy sky portion of the diagram

ATMS 8600 Dynamic ClimatologyKey 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 transmissivitythen:

where the bracketed quantity is: m total

ATMS 8600 Dynamic ClimatologyThe 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 = mclear(f)(1 - f(n))where the clear sky transmissivity is latitudinally depentent since X is latitudinally dependent.

ATMS 8600 Dynamic Climatologywhere f(n) is the % of clear sky, where f(n) represents empirically deriven cloud cover!

Clear and cloudy sky combined!And this becomes:

ATMS 8600 Dynamic ClimatologyWait for it.

ATMS 8600 Dynamic Climatology 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!!

ATMS 8600 Dynamic ClimatologyHt2 and H2s - 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)

ATMS 8600 Dynamic ClimatologyRecall 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.

ATMS 8600 Dynamic ClimatologySince 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 CO2 (96% - and 3% N2) 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.

ATMS 8600 Dynamic ClimatologyConversely 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 Ts 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.

ATMS 8600 Dynamic ClimatologyThe 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.

ATMS 8600 Dynamic ClimatologyNote: ev = f(Ts) (since water vapor is strongly correlated with surf. temp.) Atms. "holds" more water vapor as an exponential (Claussius - Clapeyron equation).andeCO2 = b1 ln(CO2) + b2 emissivity will increase as a log of increase of CO2 which means at some point you'll reach "saturation". (Oglesby and Saltzman, 1990, J. Clim.). Another reason for skepticism.

ATMS 8600 Dynamic ClimatologyThus, for 'man made' greenhouse effect' there is a point where inc. in CO2 doesn't matter anymore! The real issue is inc. in the more potent (30 time more potent, molecule for molecule) greenhouse gas: H2Ovapor.

For present-day concentrations of CO2, about 395 ppm, H2Ovapor,and O3: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!)

ATMS 8600 Dynamic ClimatologyThen with the aid of the diagram, we can write the following expressions for LW radiation:

Hs = sTs4 ( 1- ) - nnTn4(1-)

Ht = (1 - )nnTn4 + (1-n)g2T24+g1T14 + (1-n)(1-)sTs4

a further common approximation is to assume: es = en = 1.00

ATMS 8600 Dynamic ClimatologyOr that the earth's surface and clouds are taken as blackbodies. (Not valid for high cirrus = 0.8 - 0.9) atms = 0.77

The 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:

ATMS 8600 Dynamic Climatology 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:

ATMS 8600 Dynamic ClimatologyLong 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.

ATMS 8600 Dynamic ClimatologyLet'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").

ATMS 8600 Dynamic ClimatologyEven 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.

ATMS 8600 Dynamic Climatology- (1946) Similarity hypothesis, examines the lower part of the PBL as an accordion that stretches and contracts as a function of stability

ATMS 8600 Dynamic ClimatologyMonin and Obukhov also define mixing length or the rough size of eddies near the surface:

ATMS 8600 Dynamic ClimatologyWe 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

ATMS 8600 Dynamic ClimatologyFor 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:

ATMS 8600 Dynamic ClimatologyWe 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.

ATMS 8600 Dynamic Climatologyand:

where

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

ATMS 8600 Dynamic ClimatologyIf 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:

ATMS 8600 Dynamic Climatologyagain, 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.

ATMS 8600 Dynamic ClimatologyThis 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:

ATMS 8600 Dynamic ClimatologyOne 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.

ATMS 8600 Dynamic ClimatologyThe 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.

ATMS 8600 Dynamic ClimatologyWith 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.

ATMS 8600 Dynamic ClimatologyNow 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.

ATMS 8600 Dynamic ClimatologyWell we've looked at Reynold's averaging, now let's "extend" that concept:

where mm = refers to a monthly mean

now, 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.

ATMS 8600 Dynamic ClimatologyThus for temperature T we have:

and thus:

ATMS 8600 Dynamic Climatologyso:

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

ATMS 8600 Dynamic ClimatologySo we assume that typically our profiles in the EPBL look like:

ATMS 8600 Dynamic ClimatologyTemperatures 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.

ATMS 8600 Dynamic ClimatologyTo 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.

ATMS 8600 Dynamic Climatologywhat 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.

ATMS 8600 Dynamic ClimatologyNewtonian 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:

ATMS 8600 Dynamic ClimatologyHs3

which 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/cooling

This ends our look at Sensible heat flux.

ATMS 8600 Dynamic ClimatologyNow 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:

ATMS 8600 Dynamic Climatologywhere wq 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.

ATMS 8600 Dynamic ClimatologyWe could use Hydrologic Balance requirements:

This is good for large areas:

for atms:

for sfc:

ATMS 8600 Dynamic Climatologyso 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.

ATMS 8600 Dynamic ClimatologyThus 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

ATMS 8600 Dynamic Climatologyand 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:

ATMS 8600 Dynamic ClimatologyAnd, continued

then

ATMS 8600 Dynamic ClimatologyA 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:

ATMS 8600 Dynamic ClimatologyThen

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!!

ATMS 8600 Dynamic ClimatologyThen 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!

ATMS 8600 Dynamic ClimatologyBut 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

ATMS 8600 Dynamic ClimatologyThen plug that into equation for Hs4:

then manipulate and get a ratio of sens to LH, which is defined as the Bowen ratio:

ATMS 8600 Dynamic ClimatologyWhat 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

ATMS 8600 Dynamic ClimatologyBut 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) :

ATMS 8600 Dynamic Climatologywhich 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!

ATMS 8600 Dynamic ClimatologySince we can make use of alegebraic expressions (A) and substitution (B):

The substitute and do some alegebra:

ATMS 8600 Dynamic ClimatologySolving for Hs4:

Now all quantities here are know or at least measureable!

If we assume that Hs4 is a saturated point,

ATMS 8600 Dynamic ClimatologySince 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:

ATMS 8600 Dynamic ClimatologyKt - 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 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.

ATMS 8600 Dynamic ClimatologyCalculation 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?

ATMS 8600 Dynamic ClimatologySo 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?)

ATMS 8600 Dynamic ClimatologyThus 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)

ATMS 8600 Dynamic ClimatologyThus 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 balancesAtmospheric deep oceanfluxesCO2temperatures

ATMS 8600 Dynamic ClimatologyHowever, 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):

ATMS 8600 Dynamic Climatologynote 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.

ATMS 8600 Dynamic ClimatologyIf 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'.

ATMS 8600 Dynamic ClimatologyEquations:

ATMS 8600 Dynamic ClimatologyNow 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:

ATMS 8600 Dynamic Climatologythat 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.

ATMS 8600 Dynamic ClimatologyThus 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

ATMS 8600 Dynamic Climatology we can calculate OK right now,

were 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.

ATMS 8600 Dynamic ClimatologyModels 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!!

ATMS 8600 Dynamic ClimatologyPositive 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 CO2 changes postulated to arise in response to variations in mean ocean temperatures. Cooler ocean hold more CO2, warmer hold less, thus, it's a positive feedbackATMS 8600 Dynamic ClimatologyNegative 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.

ATMS 8600 Dynamic ClimatologySaltzman 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.

ATMS 8600 Dynamic ClimatologyHere's the dynamical system:

ATMS 8600 Dynamic ClimatologyThis is an unforced system. they assumed further that: a2 = a3 = bo = b2 = b3 = c1 = 0

essentially performing an ad-hoc sensitivity analysis.

Note that I, u and Tw are assumed to be departures from some equilibrium.

Final forms:

ATMS 8600 Dynamic ClimatologyNow 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

ATMS 8600 Dynamic ClimatologyInternal forcing gets cycles - Milankovitch locks it in.

2.Forced, non linear models orbital forcing is necessary and suffient to force ice age cycle

Single variable: Imbrie and Imbrie, is elegant but wrong, see Science, 207, 1980, 943 - 953.

volume of ice equation:

F = ext. forcing

ATMS 8600 Dynamic ClimatologyF = ext. forcing

I 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.

ATMS 8600 Dynamic ClimatologyThey 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