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Anthony R. Lupo 302 E ABNR Building Department of Soil, Environmental, and Atmospheric Science University of Missouri Columbia, MO 65211 Phone: 573-884-1638.

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Presentation on theme: "Anthony R. Lupo 302 E ABNR Building Department of Soil, Environmental, and Atmospheric Science University of Missouri Columbia, MO 65211 Phone: 573-884-1638."— Presentation transcript:

1 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.eduLupoA@missouri.edu Web.missouri.edu/~lupoa/atms8600.html or atms8600.pptx

2 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.

3 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!

4 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.

5 Recall in meteorology, we tend to divide phenomena by scale based on what processes are important to driving them. Table 1 ScaleTimeSpaceForceWeather Planetary7 – 14 days6000+ kmCo, PGFJet stream Synoptic1 – 7 days2000 – 6000 kmCo, PGF, Fric, Bouy Low pressure Meso1 - 24 h10 – 2000 kmBouy, PGF, FricFronts Micro< 1 hr< 10 kmBouy, Fric, PGFThunderstorm

6 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.

7 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).

8 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

9 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.

10 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)

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

12 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.

13 Another view of the climate system

14 Another view

15 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.

16 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.

17 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)

18 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)?)

19 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:

20 An idealized Temperature Spectrum for Earth

21 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.

22 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.

23 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).

24 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').

25 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!).

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

27 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?)

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

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

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

31 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;

32 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.

33 For climate: fast response (diagnostic)  atmosphere slow response (prognostic)  ice sheets, ocean (for example, if ice sheets grow, we know how atmosphere will behave)

34 Diagnostic or prognostic?

35 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".

36 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.

37 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)

38 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!

39 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)

40 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.

41 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.

42 -- 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.

43 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

44 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!

45 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.

46 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.

47 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.

48 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.

49 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: Notationc = carrier fluid a = atms. w = ocean i = ice j = trace constituents

50 Xc = 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

51 Also, 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?

52 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!

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

54 trace continuity equation Vc = velocity of carrier Vj = velocity of trace Sc = sources sinks of carrier Sj = sources sinks of trace

55 Recall from vector calculus we can put into advective forms As in atmospheric version

56 Also, we can decompose Vj as follows where, mj = molecular diffusion wjk - fall due to potential gravity field then,

57 Where, ABC A = turbulent eddy flux B = mean molecular diffusion flux C = mean sedimentation

58 Again, water vapor: j = H 2 Ov 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.

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

60 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

61 In component form (spherical coords)

62 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

63 dh (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

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

65 Take total differential of  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:

66 We can further conceptually evaluate the diabatic heating (source/ sink) – in climate these processes are of critical importance! Hrad  radiational heating H cond  conductional heating H conv  convective heating (Hcond + Hconv) = H sens H 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).

67 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. Componentform of du Dry air H2O v

68 Componentform of du Dry air + moist. Liquid water: Ice Lithosphere

69 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)

70 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)

71  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):

72 (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

73 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!!!

74 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!!

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

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

77 Recall concept of Reynolds Averaging Where X’ is much smaller than X bar Instantaneous 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!

78 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!

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

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

81 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!)

82 What 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.

83 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 (considered no change for Gen Circ.): (dry atms. + water)

84 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

85 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;

86 assume atmosphere is well behaved, reverse order of integration and differentiation (and assume w t = 0): AB C we call dry mass integral (also a,b,c are boundary recall)

87 Where A = Area integral (in latitude (  ) and longitude ( )) 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

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

89 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!);

90 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.

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

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

93 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!

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

95 Then for water vapor equation: horiz and vert cloud surf vapor flx formation Redefine Xv (mass of water vapor m v ):

96 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)

97 Diagram

98 Then (Atmospheric water vapor): precipitablehorizontal Evap precip Cloud watertransport water Using the same procedures as before, we change variables and handwave.

99 Cloud water: transport ofprecip phase clouds changes Surface Water (Soil science folks like this): Evap Meltingliquid precip

100 Distinguish 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!

101 Surface Ice (fundametnal cont. equation of glaciology) snowsnowsnow evapmelt 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!

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

103 Then add 'em all up to show balance: Conservation for water in climate sytem! In Practice: m w >> m i >> m v + m n and m v >> m n m w = 1.400 x 10 21 kg m i = 2.0 x 10 19 kg and m v + m n = 1.6 x 10 16 kg

104 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

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

106 Angular 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:

107 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:

108 globally integrate: Where:  = mean angular velocity of the Earth I = Moment of Inertia

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

110 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  0 What are each of the terms?

111 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!

112 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 Viron 1,2,* andJ. O. Dickey 3, 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/http://www.aoml.noaa.gov/general/enso_faq/

113 Term (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

114 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)- 

115 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).

116 Recall Atms 8400 equation:

117 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).

118 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)

119 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.

120 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).

121 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

122 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.

123 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.

124 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.

125 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.

126 term (1)  is the “pressure work” term term (2)  dissipation (friction) term (3)  radioactivity (Uranium, etc.), chemical, geothermal term (4)  diabatic heating

127 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)

128 Internal energy per unit mass: and Expressions for e: Component Internal EnergyPhase Changes Atms (a) vapor freezing

129 Component Internal Energy Phase Changes Ocean (w,i) Sea ice Land (l) Freezing of ground and/or lakes

130 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).

131 We can decompose: vaporice These are Fluxes of Latent heat of vaporization and ice which can be decomposed into:

132 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:

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

134 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!

135 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 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.

136 Remember This?

137 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

138 Bottom boundary (Atmosphere): Integrate:

139 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.

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

141 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):

142 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 Snowmelt In the above expression: H1 = Shortwave in (solar, space) H2 = Shortwave out H3 = Sensible heating (conduction and convection)

143 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):

144 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 Snowmelt In the above expression: H1 = Shortwave in H2 = Shortwave out H3 = Sensible heating (conduction and convection)

145 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.

146 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:

147 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.)

148 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)

149 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!

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

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

152 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 =  and IE = c v T

153 PE + IE = TPE =  c p T Now put back into our equation for energy: Total Energy In the Atmospheric Column!!!!

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

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

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

157 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.

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

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

160 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.

161 You can substitute for Pi using: L f q i and then substitute continuity for a glacier (fundamental equation of glaciology):

162 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!

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

164 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

165 Then you are left with (heat fluxes in = heat fluxes out): In a more practical sense: we really want to take the limit: where  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. H D 2 and H D 1 disappear.

166 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.

167 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!!!

168 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!!!

169 The equation:

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

171 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.

172 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.

173 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.

174 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.

175 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

176 rt - reflectivity above clouds rs - surface reflectivity (albedo!) X Opacity / absorptivity of atmosphere (up and down) Ht(1) - SW in is equal to R!

177 Schematic (clear sky):

178 Cloudy Sky Diagram:

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

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

181 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( ,t)  solar radiation on a horizontal surface at the top of the atmosphere. X = X( ,t)  shortwave absorptivity, which is a function of zenith angle, which determines a path length. r = r( ,t)  shortwave clear sky reflectivity r = rs(  )  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.

182 rt = rt(  )  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(  )  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

183 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

184 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: X d (0 o )  value of absorptivity at the equator P(  )  path length as a function of latitude P  = R(0) at equinox / R(  ) at equinox or “normalization factor”.

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

186 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!

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

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

189 The actual values of  clear,  cloudy, and  total are not well known! Many studies approximate as;  total = (1-X) (1-r) this is similar to the more complex expression, while others have used the following:  =  clear(  )(1 - f(n)) where the clear sky transmissivity is latitudinally depentent since X is latitudinally dependent.

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

191 Wait for it.

192  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!!

193 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)

194 Recall from R-T, that emissivity (  ) = 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  or 'emissivity'. Recall that all things emit radiation proportional to T to the 4th power, Stefan Boltzmann's law.

195 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 (  )'s. Absorption and re-emission of LW radiation is dependent on "path length" of gasses which is dependent on atmospheric density. 2 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.

196 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% CO 2, 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  = 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.

197 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.

198 Note:  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  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….

199 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 :  = 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!)

200 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:  s =  n = 1.00

201 Or 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  = 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:

202 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:

203 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.

204 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").

205 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.

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

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

208 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

209 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:

210 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.

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

212 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:

213 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.

214 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:

215 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.

216 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.

217 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.

218 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.

219 Well 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.

220 Thus for temperature T we have: and thus:

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

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

223 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.

224 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.

225 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.

226 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:

227 Hs3 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.

228 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:

229 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.

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

231 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.

232 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

233 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:

234 And, continued then

235 A frequent and very critical assumption that is made by all climate models is: or that Drag. Coefficient for H 2 0Vapor = 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:

236 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!!

237 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 H 2 O vapor?) Theoretically, real evaporation = potential evaporation over the oceans!

238 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

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

240 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

241 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) :

242 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!

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

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

245 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:

246 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.

247 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)

248 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!

249 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

250 And away we go.

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

252 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.

253 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 HT 2 (long wave out) as part of the energy source for the climate system). When HT 1 and HT 2 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.

254 We're familiar with energy balance and implied transports:

255 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?

256 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!

257 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:

258 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:

259 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:

260 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!

261 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.

262 Thus: so: Now with Q  340 W/m 2 and assuming a present day value of Ap  0.3, Teff  255 K

263 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?

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

265 these empirically take into account the global effects of green house gasses, and H 2 0 and CO 2. Budyko found these based on real data and independent of temperature. Most important greenhouse gas is water vapor. Then equating Ht 1 and Ht 2 (we now have a very simple EBM!)

266 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!

267 Here's the functional representation:

268 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)

269 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!!!).

270 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!!!

271 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):

272 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

273 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.

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

275 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,

276 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.

277 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.

278 Let's try Budyko's parameterization: Use, with a = 203 W/m 2 and b = 2.09 W/m2 K, then B = 1.12 K. So a change in So of 14 W/m 2 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/m 2 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!

279 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/m 2, thus, as much as half the "Global Warming" might be solar variability. As a final note, we can write: and let:

280 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.

281 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).

282 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:

283 1)short wave radiation f(  ) 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.

284 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!

285 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!!!!

286 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.

287 Budyko and Sellers in 1968 came up with independently derived parameterizations: Budyko's: ZonalGlobal AvgAvgT’s This 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.

288 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!!!!)

289 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:

290 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  i at any desired latitude.

291 That is: using T s (i) = -10 C, A p (i) = 0.47 and a p (i) = 0.53 and a and b are as before.

292 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!!

293 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.

294 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

295 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

296 2.F = v'T' heat flux poorly parameterized in terms of Ts a.should depend on atmospheric T not Ts, or more importantly temp gradients (hemispheric asymmetry) b.no latent heat flux c.no energy flux due to mean motions. 3. should depend on atmospheric state (clouds) as well as surface state 4.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.

297 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!

298 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.

299 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!

300 Schematically our domain again

301 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.

302 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.

303 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

304 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 xy 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)

305 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: u o u 1

306 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).

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

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

309 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.

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

311 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.

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

313 V - eqn

314 Hydrostatic balance: Thermodynamics:

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

316 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!!!

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

318 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).

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

320 5)  = 0 at p = ps and p = 0 (rigid - lid assumption). 6)Treat the SBL as a separate layer: and where we assume C D = C D o 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:

321 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:

322 c.  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.   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).

323 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.

324 Summarize these with the following expressions;

325 and finally

326 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).

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

328 Thermal Wind Ekmann Balance in PBL: geostrophy fric

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

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

331 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:

332 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:

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

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

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

336 kl is constant for land and ice kw is for water Atmospheric:

337 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

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

339 Prescribe 5 surface types: bare land, open water, snow continental ice, and sea ice. Ntypekw rs()rs() 1Oceank*-c 6 (T s -T D )1.0 a(  ) 2Sea ice50.5 b(  ) 3Land10.80.2 4Snow10.5 b(  ) 5Continental ice10.5 b(  )

340 Now prescribe or parameterize the stress terms First, 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.

341 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.

342 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”

343 where A is a model creation horizontal gradient lapse rate Eddy water vapor transport: used simple mixing length argument – considerations these are quite plausible in this case!

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

345 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.

346 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.

347 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

348 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'.

349 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).

350 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.

351 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.

352 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.

353 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.

354 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.

355 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.

356 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.

357 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.

358 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.

359 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.

360 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)

361 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.

362 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

363 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.

364 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.

365 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.

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

367 And where; And;

368 Then as before: Total Energy Now integrate this globally: area of earth

369 when we perform these operations we get the beastly equation: net radiation inwater mass phase changesgeothermal surface topography where:

370 Let's make a few reasonable assumptions to aid in our scale analysis: Ma = const Ml = Const Mw + Mi = Const or that:

371 Now divide through by the area of earth to change from W (Watts or total global power) to W/m 2 (globally averaged value) to get another beastly equation:

372 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 Areasdensities  = 5 x 10 14 m 2  a = 1 kg m -3  w = 5 x 10 14 m 2  w = 1000 kg m -3  l = 5 x 10 14 m 2  i = 920 kg m -3  i = 5 x 10 14 m 2 (at T = 275 K)

373 Mass of each where seasonal cycle is important: Ma = 5 x 10 18 kg Mw = 1.4 x 10 21 kg Ml = 1.6 x 10 18 kg Mi = 26 x 10 18 kg (now) Mi = 77 x 10 18 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

374  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).

375 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.

376 Thus how do we evaluate: For our ICE Age Pleistocene situation, a reasonable guess is that 50 x 10 18 kg of ice accumulated / melted in about 10,000 years. Then: Joules of energy! (sun outputs 3.9 x 10^26 J/s).

377 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

378 = 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:

379 = 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 = ? determine from scale analysis, We'll keep this ???? for now since ocean is 3 orders of magnitude bigger than other components of climate system.

380 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 (  )? our main source of changing  will be elevation of the ice sheets themselves. ow, let's re-write our scaled equation, listing characteristics values for each term:

381 Now, let's re-write our scaled equation, listing characteristics values for each term: 10-310-410-3???? atms T change land T change ice T changeocean T change 10-3?????10-1 dry static energynet radiative fluxgeothermal

382 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.

383 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).

384 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"?

385 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

386 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.

387 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!

388 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.

389 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?

390 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?)

391 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)

392 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 balancesAtmospheric deep ocean fluxesCO2temperatures

393 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):

394 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.

395 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'.

396 Equations:

397 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:

398 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.

399 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

400 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.

401 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!!

402 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

403 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.

404 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.

405 Here's the dynamical system:

406 This 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:

407 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

408 Internal 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

409 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.

410 So they rewrite: if F >= I decay if F <= I build-up

411 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.

412 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.

413 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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