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*Notes for climate dynamics course (EPS 231) 2.1 Multiple equilibria and climate stability 1. energy*

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Notes for climate dynamics course (EPS 231)

Eli Tziperman

April 21, 2011

Contents 1 Introduction 2

2 Basics, energy balance, multiple climate equilibria 2 2.1 Multiple equilibria and climate stability . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Small ice cap instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

3 ENSO 4 3.1 ENSO background and delay oscillator models . . . . . . . . . . . . . . . . . . . 4 3.2 ENSO’s irregularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.2.1 Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2.2 Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.3 Teleconnections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4 Thermohaline circulation 7 4.1 Preliminaries, scaling, energetics, Stommel box model . . . . . . . . . . . . . . . 8 4.2 Stability and multiple equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.3 Advective and convective feedbacks . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.4 THC variability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4.4.1 (time permitting) More on stochastic THC variability . . . . . . . . . . . . 10

5 Dansgaard-Oeschger, Heinrich events 11

6 Glacial cycles 12 6.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 6.2 Glacial cycle mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 6.3 Milankovitch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 6.4 CO2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

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7 Equable climate 14 7.1 Equator to pole Hadley cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 7.2 Polar stratospheric clouds (PSCs) . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 7.3 Hurricanes and ocean mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 7.4 Convective cloud feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1 Introduction This is an evolving detailed syllabus of EPS 231, see course web page, all course materials are available under the downloads directory.

Homework assignments (quasi-weekly) are 50% of final grade, and a final course project con- stitutes the remaining 50%. There is an option to take this course as a pass/fail with approval of instructor.

2 Basics, energy balance, multiple climate equilibria Downloads available here.

2.1 Multiple equilibria and climate stability 1. energy balance 0d.pdf with the graphical solutions of the steady state solution to the

equation CTt = (Q/4)(1−α(T )− εσT 4 obtained using energy balance 0d.m, and then the quicktime animation of the bifurcation behavior.

2. Some nonlinear dynamics background: saddle node bifurcation ((p 45, Strogatz, 1994) or Applied Math 203 notes p 47), and then the energy balance model as two back to back saddle nodes and the resulting hysteresis as the insolation is varied;

3. Climate implications: (1) faint young sun paradox! (2) snowball (snowball obs from Ed Boyle’s lecture);

2.2 Small ice cap instability 1. Introduction: The Budyko and Sellers 1d models Simple diffusive energy balance models

produce an abrupt disappearance of polar ice as the global climate gradually warms, and a corresponding hysteresis. The SICI eliminates polar ice caps smaller than a critical size (18 deg from pole) determined by heat diffusion and radiative damping parameters. Below this size the ice cap is incapable of determining its own climate which then becomes dominated, instead, by heat transport from surrounding regions. (Above wording from Winton (2006), teaching notes below based on North et al. (1981)).

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http://www.seas.harvard.edu/climate/eli/Courses/EPS231/2011spring/ http://www.Seas.Harvard.edu/climate/eli/Courses/EPS231/Sources/ http://www.Seas.Harvard.edu/climate/eli/Courses/EPS231/Sources/02-Basics http://www.seas.harvard.edu/climate/eli/Courses/EPS231/Sources/02-Basics/energy_balance_0d.m http://www.seas.harvard.edu/climate/eli/Courses/EPS231/Sources/02-Basics/lecture_bif1d1_eli.pdf http://www.seas.harvard.edu/climate/eli/Courses/EPS231/Sources/02-Basics/Boyle-snowball-MIT12.842.pdf

2. Derive the 1d energy balance model for the diffusive and Budyko versions (eqns 22 and 33): Assume the ice cap extends where the temperature is less than -10C, the ice-free areas have an absorption (one minus albedo) of a(x) = a f , and in the ice-covered areas a(x) = ai; ap- proximate the latitudinal structure of the annual mean insolation as function of latitude using S(x) = 1+S2P2(x), with P2(x) = (3x2−1)/3 being the second Legendre polynomial (HW); add the transport term represented by diffusion (r2 cosθ)−1∂/∂θ(Dcosθ∂T/∂θ), which, us- ing x = sinθ, 1− x2 = cos2 θ and d/dx = (1/cosθ)d/dθ gives the result in the final steady state 1D equation (22),

− d dx

D(1− x2)dT (x) dx

+A+BT (x) = QS(x)a(x,x0).

as boundary condition, use the symmetry condition that dT/dx = 0 at the equator, leading to only the even Legendre polynomials.

3. (time permitting) Alternatively to the above, we could model the transport term a-la Budyko as γ[T (x)−T0] with T0 being the temperature averaged over all latitudes, and the temperature then can be solved analytically (HW).

4. How steady state solution is calculated: eqns 22-29; then 15 and 37, the equation just before 37 and the 4 lines in the paragraph before these equations. Result is Fig. 8: plot of Q (solar intensity) as function of xs (edge of ice cap). Analysis of results: see highlighted Fig. 8 in sources directory, unstable small ice cap (which cannot sustain its own climate against heat diffusion from mid-latitudes), unstable very-large ice cap (which is too efficient at creating its own cold global climate and grows to a snowball), and stable mid-size cap (where we are now).

5. Heuristic explanation of SICI: Compare Figs 6 and 8 in North et al. (1981), SICI appears only when diffusion is present. It is therefore due to the above mentioned mechanism: competi- tion between diffusion and radiation. To find the scale of the small cap: it survives as long as the radiative effect dominates diffusion: BT∼DT/L2, implying that L∼

√ D/B. Units:

[D] =watts/(degree K×m2) (page 96), [B] =m2/sec (p 93), so that [L] is non dimensional. That’s fine because it is in units of sine of latitude. Size comes out around 20 degrees from pole, roughly size of present-day sea ice!

6. The important lesson(!): p 95 in North et al. (1981), left column second paragraph, apolo- gizing for the (correct...) prediction of a snowball state.

7. This is a complex PDE (infinite number of degrees of freedom), displaying a simple bifur- cation structure. In such case we are guaranteed by the central manifold and normal form theorems that it can be transformed to the normal form of a saddle node near the appro- priate place in parameter space. First, transformed to center manifold and get an equation independent of stable and unstable manifolds (first page of lecture 04 cntr mnfld.pdf); next, transform to normal form within center manifold (lecture 03 bif1d2.pdf, p 89).

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8. (time permitting) Numerically calculated hysteresis in 1D Budyko and Sellers models: fig- ures ebm1d-budyko.jpg and ebm1d-sellers.jpg obtained using ebm 1dm.m).

9. (time permitting) One noteworthy difference between Budyko and Sellers is the transient behavior, with Budyko damping all scales at the same rate, and Sellers being scale-selective. (original references are Budyko, 1969; Sellers, 1969).

3 ENSO Downloads available here.

3.1 ENSO background and delay oscillator models Sources: Woods Hole (WH) notes (Cessi et al., 2001), lectures 0, 1, 2, here, plus the following: Gill’s atmospheric model solution from Dijkstra (2000) technical box 7.2 p 347; recharge oscillator from Jin (1997) (section 2, possibly also section 3);

• The climatological background: easterlies, walker circulation, warm pool and cold tongue, thermocline slope (lecture 1 from WH notes).

• Dynamical basics: equatorial Rossby and Kelvin waves, thermocline slope, SST dynamics, atmospheric heating and wind response to SST from Gill’s model (rest of WH lecture 1). The coupled feedback,

• The heuristic delayed oscillator equation from section 2.1 in WH notes. One detail to note regarding how do we transition from +b̂hoff−eq(t− [12τR+τK]) to −b̄τeq(t− [

1 2τR+τK]) and

then to −bT (t− [12τR + τK]): hoff−eq depends on the Ekman pumping off the equator. In the northern hemisphere, if the wind curl is positive, the Ekman pumping is positive, upward (wE = curl(~τ/ f )/ρ), and the induced thermocline depth anomaly is therefore negative (a shallowing signal). The wind curl may be approximated in terms of the equatorial wind only (larger than the off-equatorial wind), consider the northern hemisphere:

hoff−eq ∝−wEkmanoff−eq ∝−curl(τoff−eq)≈ ∂yτ (x) off−eq ≈ (τ

(x) off−eq− τ

(x) eq )/L ∝−τ

(x) eq .

Finally, as the east Pacific temperature is increasing, the wind anomaly in the central Pacific is westerly (positive), leading to the minus sign in front of the T term,

−τ(x)eq (t− [ 1 2

τR + τK])/L ∝−T (t− [ 1 2

τR + τK]).

• Next, the linearized stability analysis of the Schopf-Suarez delayed oscillator from the WH notes section 2.1.1. Show