The Energy Balance Model

1 The Energy Balance Model 2 D.S. Battisti 3 Dept. of Atmospheric Sciences, University of Washington, Seattle Generated using v.3.2 of the AMS LATEX template 1

ABSTRACT 5 ad 2

6 1. Zero-order climatological energy balance model without an absorbing atmosphere 7 8 Assume the atmosphere is transparent to the insolation (mainly visible and ultraviolet) and that the energy reaching the planet averaged over a day is S = S o Λ(x,t), (1) 9 1 11 12 13 1 15 where S o (the solar constant) depends on solar luminosity and distance from the sun and is 137 W/m 2 today. Λ(x,t) takes into account the angle of incidence as a function of latitude which depends on time of day, time of year, and Milankovich components eccentricity, precession and obliquity. Figure 1 shows the climatological annual cycle of S for Earth, assuming present day orbital parameters. In the simplest case where all energy arriving at the planet s surface is absorbed, the total absorbed energy is S o πr 2, (2) 16 17 18 19 where r is the radius of the planet. In general, a fraction of the incident energy is reflected back to space. This fraction is called the albedo (α). The fraction of the incident solar energy that is reflected back to space by the aggregate composition of the climate system is called the planetary albedo α p. Hence, the net energy absorbed by a planet is S o πr 2 (1 α p ). (3) 2 21 In equilibrium, the net energy absorbed must be balanced by the net radiation emitted to space. Assuming that the planet is a perfect blackbody, the radiation to space is σt e (πr 2 ), () 3

22 23 where T e is the emitting temperature of the planet (annual, globally averaged) and σ is the Stephan- Boltzman constant, σ = 5.67 1 8 Wm 2 K 1. An energy balance therefore requires S o (1 α p )/ = σt e. (5) 2 a. Application to Earth 25 26 27 28 29 Earth has a planetary albedo of.3, of which.2 is due to reflections from objects in the atmosphere (mainly clouds) and.6 from objects on the surface (mainly ice and snow) (Fig. 2). The annual averaged insolation reaching the top of the atmosphere S o is 137 Wm 2. Plugging these numbers into Eq. (5), we find T e = 255 K. The observed surface annual and global averaged surface temperature is 288 K a difference of 33 K. 3 2. Energy balance in an absorbing atmosphere 31 32 33 Approximate the atmosphere as a thin radiating slab with temperature T a and with a selection of greenhouse gases such at the net emission is some fraction ε of that of a blackbody at the same temperature. The net energy emission from the atmosphere is therefore εσt a (πr 2 ), (6) 3 35 36 where ε is some bulk emissivity. Taking into account the fact that the slab atmosphere will radiate to both to space and back down to the ground, we can now write the full energy balance for the surface temperature T s for a more realistic earth with greenhouse gases: S o (1 α) (πr 2 ) + εσt a (πr 2 ) = σt s (πr 2 ) (7) 37 or S o (1 α)/ + εσt a = σt s. (8)

38 39 The atmosphere in our model gets no direct energy from the Sun, and recognizing that partial emitters are partial absorbers, the atmosphere gets only some of the energy emitted by the ground, so that the energy balance for the atmosphere requires that 2εσT a = εσt s. (9) 1 With no circulation, the solution to Eqs. (8) and (9) is therefore given by T s = S o (1 α) σ(1 ε/2), T a = T s / 2. (1) 2 3 With no absorbing atmosphere (ε = ), the surface temperature T s would be 255K, or 18 C. The observed surface temperature T s = 288 K is obtained using a bulk emissivity ε =.76 and Eq. (8) yield T a = 22 K. 1 5 3. Adding circulation 6 7 8 We can create a simple zonally average energy balance model by assuming our EBM holds locally. We can also add a simple treatment of the heat flux convergence D by atmospheric circu- lation to the steady state the atmospheric energy budget: 2εσT a = εσt s D, (11) 9 5 51 52 where D is the convergence of energy per unit area due to circulation, and the energy balance at the ground is given by Eq. (8). Applied to the zonal average climate, D represents the convergence of energy associated with the meridional energy transport (Fig. 3). The solution to Eqs. (11) and (8) using an idealized approximation to the observed D is shown in Fig.. 1 To be more precise, the global average temperature will be the global average of the local T e, calculated by replacing S o in Eq. (5) by S o Λ, where ( ) denotes the climatological average. This yields T e = 25 K and T s = 286 K. 5

53. The time dependent Energy Budget Model 5 We can make the EBM time dependent as follows: 55 C a T a t C g T s t = 2εσT a + εσt s D (12) = S(1 α)/ + εσt a σt s, (13) 56 57 58 where S = S o Λ and the left hand sides show the rate of change of energy storage in the atmosphere (Eq. (12)) and surface (Eq. (13)) (treating surface as either ocean or dirt). To simplify, these equations can be linearized about a reference temperature T r = 273.13K: T a = T r + T a T s = T r + T s (1) 59 where T a /T r, T a /T r << 1. Equations (12) and (13) then become C a T a t C g T s t = a ε + b ε T s 2 b ε T a D (15) = S(1 α)/ a (1 ε) + ε b T a b T s, (16) 6 where we have dropped the primes and the constants a and b are a = σt r ; b = a/t r. (17) 61 62 Plugging in numbers, we find a = 316 Wm 2 and b =.6 Wm 2 K 1. The heat capacity of the atmosphere is C a = C p a P g = (1 3 J kg 1 K 1 ) (1 kg m 2 ) = 1 7 J m 2 K 1. (18) 63 6 65 Let s first assume the ground is ocean, and we are interested in the seasonal cycle of tempera- ture. In this case, only the near surface ocean (basically the mixed layer) participates and we can estimate the heat capacity of the ground to be C g = C o = C p o h ρ H2 O, (19) 6

66 where h is the depth of the mixed layer. If we assume the mixed layer depth h = 75m, then we find C g = C o = ( 1 3 ) (75) (1 3 ) = 3 1 8 Jm 2 K 1. (2) 67 68 69 Hence, over the water covered planet, C g /C a = C o /C a 3 >> 1 and thus we can assume the atmosphere is in equilibrium with respect to changes in the surface (sea surface) temperature. This allows us to set the left-hand side of Eq. (15) to zero and obtain: = aε + bεt s 2bεT a D (21) 7 71 and the equation for the surface temperature Eq. (16) (that is, the sea surface temperature) simpli- fies to C o T s t = S (1 α)/ A B T s + D/2, (22) 72 73 7 75 76 where A = a (1 ε/2) = 195 Wm 2 and B = b (1 ε/2) = 2.9 Wm 2 K 1. The equilibrium global average (D = ) solution to Eq. (21) and (22) is T s = ( 137(1.3) ) 195 /2.9 = 15. C, T a = 26.6 C (23) and is independent of the heat capacity of the atmosphere and surface. Eq. (23) is the linearized version of Eq. (1). It is useful to rearrange Eq. (22) and define the forcing F to be F S(1 α)/ A = C o T s t + B T s, (2) 77 which has the time dependent solution T s = e t/τ t F(t) C o e t/τ dt, (25) 78 where τ = C o /B (26) 7

79 8 is the characteristic response (adjustment) time of the system. For a planet covered by 75m of water, this is about τ = C o /B = 3. x 18 2.9 s = 3.3 years. (27) 81 By way of contrast, the heat capacity associated with seasonal time scales over land is C g = C l = C p dirt h dirt ρ dirt = 12J kg 1 K 1 1m 25 kg m 3 = 3 1 6 J m 2 K 1. (28) 82 83 8 For a land covered planet, the adjustment time scale is < 2 weeks. For an instantaneous switch-on of a constant forcing, t F = F o t > the solution is (29) T s = F o B (1 e t/τ ). (3) 85 5. A simple model of the Seasonal Cycle 86 Subtracting the long term mean from Eq. (22), we find C o T s t = S (1 α)/ B T s, (31) 87 where prime denotes the deviation from the long term mean. Expanding Eq. (31) in a Fourier 88 series S T s = j S j T j exp(i ω jt), (32) 89 the solution is C o i ω j T g j = S j (1 α)/ BT g j, (33) 8

9 91 where or, if S j is real, T j = S j(1 α)/ B + ic o ω j, (3) { } T j = Re T j e (iω jt) = S j (1 α)/ [ B 2 + (C o ω j ) 2] 1/2 cos(ω j t φ j ), (35) 92 where φ j = tan 1 ( C o ω j /B ). (36) 93 a. Annual cycle of Sea Surface Temperature (SST) 9 For the annual harmonic and over the ocean (ω = 2π/(π x 1 7 ) s 1, C o = 3 1 8 ), so φ j π/2 (37) 95 96 or φ j = 3 months. In the midlatitudes, the forcing S j / 2 Wm 2 (see Fig 1) and so from Eq. (3) we find T j 2 C. (38) 97 98 99 1 11 12 13 1 15 So the annual cycle in the midlatitude oceans should have amplitude ±2 C, and lag the maximum in insolation by about 3 months. The observed seasonal harmonic of SST (zonally averaged) is shown in Fig. 7. The solution to Eq. (3), linearized about the equilibrium solution with heat transport (Eq. (11), is shown in Fig. 5. A more realistic model would take into account general latitudinal dependence of mixed layer depth deepening in the high latitudes due to vigorous mixing be atmospheric winds and increased buoyancy loss due to wintertime cooling; such a calculation is shown in Fig. 6. In the midlatitudes, this is a fair model of the annual cycle of SST in the Southern Hemisphere, but it underestimates by a factor of three the seasonal cycle in the Northern Hemisphere (Fig. 7). 9

16 17 Large discrepancies also appear near the equator illuminating the zero order impact of ocean dynamics on the annual cycle of SST along the equator. 18 b. Seasonal Cycle over land 19 11 111 112 113 The seasonal cycle of temperature over land is much greater than the seasonal cycle of SST mainly because the mass of land that participates is much less than the mass of the ocean because diffusion through a solid is a much less efficient process than mixing of a fluid. Assuming that the heat capacity of the soil that participates in the annual cycle is small compared to the heat capacity of the atmosphere, the equations 15 and 16 are (after removing the time mean climatology) C a T a t = +b ε T s 2 b ε T a (39) = S(1 α)/ + ε b T a b T s, () 11 which has the solution Fourier solution: T a j = T s j = S j (1 α)/ 2B + ic a ω j /ε (1) 1 bε (iσc a + 2bε)T a j, (2) 115 116 117 118 119 Plugging in numbers, we find the amplitude of the annual cycle of surface temperature over midlatitude land regions to be in excess of ± C and lag insolation by 8 days or so. This is too extreme largely because we have assumed zero heat capacity. Using a more realistic heat capacity yields the amplitude of the annual cycle of surface temperature of ±3 C and a lag of 3 days. 1

12 6. What is missing? 121 122 123 12 125 126 127 128 129 13 Lots of things. But the three most important are (i) the atmosphere absorbs some ( 18%) of the incident solar radiation mainly by ozone in the stratosphere and water vapor in the troposphere; (ii) turbulent exchange of sensible and latent heat at the surface and (iii) winds advect temperature. The first term is fundamental, albeit unappreciated (the seasonal cycle of temperature in the troposphere is mainly due to absorption of solar energy in the troposphere; see Donohoe and Battisti (213)). The second term mainly acts to amplify (damp) the seasonal cycle in the atmosphere (land) over the land regions, and acts to damp (amplify) the seasonal cycle in the atmosphere (land) over the ocean regions. The final term is also important, as in the midlatitudes westerly winds cause a blending of two fundamentally different end-member solutions discussed in section 5 (see also Donohoe and Battisti (213);?). 11

131 References 132 133 Donohoe, A., and D. S. Battisti, 211: Atmospheric and surface contributions to planetary albedo. Journal of Climate, 2 (16), 2 18. 13 135 Donohoe, A., and D. S. Battisti, 213: The seasonal cycle of atmospheric heating and temperature. Journal of Climate, 26 (1), 962 98. 136 137 138 McKinnon, K. A., A. R. Stine, and P. Huybers, 213: The spatial structure of the annual cycle in surface temperature: Amplitude, phase, and lagrangian history. Journal of Climate, 26 (2), 7852 7862. 139 1 Trenberth, K. E., and J. M. Caron, 21: Estimates of meridional atmosphere and ocean heat transports. Journal of Climate, 1 (16), 333 33. 12

11 12 LIST OF FIGURES Fig. 1. The top of the atmosphere incoming insolation.............. 1 13 1 15 16 17 18 19 15 151 152 153 15 155 156 157 158 159 16 161 162 163 16 Fig. 2. Fig. 3. Fig.. Fig. 5. Fig. 6. (a) Surface and (c) planetary albedo, and (b) surface and (d) atmospheric contributions to (c). All quantities are expressed as a percentage, where 1% corresponds to.1 units of albedo. From Donohoe and Battisti (211)............... 15 The climatological annual average northward energy transport. The solid line is the total transport by the ocean and atmosphere, deduced from the top of the atmosphere net radiation and with the assumption that the system is in equilibrium. The dashed lines are estimates of the atmospheric contribution to the total transport, calculated from two reanalysis products. Due to uncertainty in ocean data, the ocean contribution is most reliably estimate by the difference between the the total transport and the atmospheric transport. From Trenberth and Caron (21)...................... 16 The zonally and annual average insolation (blue curve) and solutions to the various energy balance models: (red) the emitting temperature (Eq. (5)); (yellow) with an absorbing atmosphere but no circulation (Eq. (1) with D = ); and (purple) with an absorbing atmosphere and circulation (D = Q o cos(3φ) in Eq. (11), where Q o = 5 Wm 2 and φ is latitude). The units are Wm 2 and K for insolation and temperature, respectively......... 17 Solution to the linearized time-dependent energy balance model Eq. (31), forced by the annual cycle of insolation (with the time mean removed). In this calculation, the mixed layer depth is taken to be 75m, α =.3, ε =.76, and the model has been linearized about the nonlinear solution to the equilibrium solution with idealized meridional energy transport is in Fig.. Contour interval is...,-2, -1, -.3, -.2, -.1,,.1,.2,.3, 1, 2,... C..... 18 As in Fig. 5, but including a simple parameterization of mixed layer depth as a function of latitude......................... 19 165 Fig. 7. Observed seasonal cycle zonally averaged SST. Contour interval is 2 C........ 2 13

Daily Averaged Insolation (W/m 2 ) 1 2 3 5 2 1 3 5 5 5 5 15 5 15 25 25 35 35 5 5 5 3 2 1 5 15 2535 5-5 5 5 2 6 8 1 12 Daily Insolation-Weighted FIG. 1. The top of the atmosphere Average incoming Cosine insolation. of Zenith Angle.2.2.1.1.3..5.3 5.6.7.7.6-5.5..2.1.3 2 6 8 1 12 5 3 2 1.6..2 1

15 AUGUST 211 D O N OHOE AND BATTISTI 5 166 167 168 FIG. 2. (a) Surface and (c) planetary albedo, and (b) surface and (d) atmospheric contributions to (c). All quantities are expressed as a percentage, where 1% corresponds to.1 units of albedo. FIG. 2. (a) Surface and (c) planetary albedo, and (b) surface and (d) atmospheric contributions to (c). All aquantities P,ATMOS 5areR, expressed and as a percentage, where () 1% andcorresponds a P,SURF are to reported.1 unitsin ofappendix albedo. From B. The qualitative conclusions found in this study are independent of the Donohoe and Battisti (211). a P,SURF 5 a (12R2A)2 1 2 ar. (5) assumptions made in the simplified radiative transfer model. Equation () is due to direct reflection by the atmosphere [the first term on the right-hand side of Eq. (1)]. All of the solar radiation that is reflected by the surface and eventually passes through the TOA [the second term on the right-hand side of Eq. (1)] is attributed to a P,SURF. By definition, the surface and atmospheric contributions to planetary albedo sum to the planetary albedo: a P 5 a P,ATMOS 1 a P,SURF. (6) Maps of a P,SURF and a P,ATMOS are shown in Figs. 2b,d. We calculate a P,ATMOS and a P,SURF using annual average (solar weighted) data. We have also performed the partitioning on the climatological monthly mean data and then averaged the monthly values of a P,ATMOS and a P,SURF to obtain the annual average climatology. The annual and zonal average a P,ATMOS calculated from the monthly data agree with that calculated directly from the annual average data to within 1% of a P,ATMOS at all latitudes. We note that Taylor et al. (27, hereafter T7) used a similar simplified radiative transfer model to partition planetary albedo into surface and atmospheric components. In contrast to our formulation, T7 assumed absorption only occurs on the first downward pass through 15 2) RESULTS The maps of surface and planetary albedo exhibit large values in the polar regions, with larger spatial differences in the meridional direction than in the zonal direction (Fig. 2); the predominant spatial structure in both maps is an equator-to-pole gradient. Significant meridional gradients in surface albedo are constrained to be at the transition to the cryospheric regions (around 78 in each hemisphere), whereas the meridional gradients in planetary albedo are spread more evenly across the storm-track regions (from 38 to 68). The percentage of solar radiation absorbed during a single pass through the atmosphere (not shown) features a predominant equator-to-pole gradient with tropical values of order 25% and high-latitude values of order 15% with still smaller values occurring over the highest topography. The global pattern of atmospheric solar absorption is virtually identical to the pattern of vertically integrated specific humidity [from National Centers for Environmental Prediction (NCEP) reanalysis] with a spatial correlation coefficient of.92; this is expected because the atmospheric absorption of solar radiation is predominantly due to water vapor and ozone

RTH AND CARON 335 m 2 ) 169 17 171 172 ren-17are17 rmer 986 varicted disicrotrucling oist ansthe from FIG. 2. The required total heat transport from the TOA radiation FIG. 3. The climatological annual average northward energy transport. The solid line is the total RT is given along with the estimates of the total atmospheric transport AT from NCEP and ECMWF reanalyses (PW). transport by the ocean and atmosphere, deduced from the top of the atmosphere net radiation and with the assumption that the system is in equilibrium. The dashed lines are estimates of the atmospheric contribution to the total transport, calculated from two reanalysis products. Due to uncertainty in ocean data, the ocean contribution is most reliably estimate by the difference between the the total transport fluxes are shown by Trenberth et al. (21a) to be unreliable south of about 2 N where there are fewer than 25 observations per 5 square per month. In addition, TOA biases in absorbed shortwave, outgoing longwave, and net radiation from both reanalysis NWP models are substantial ( 2 W m 2 in the Tropics) and indicate that clouds are a primary source of problems in the NWP model fluxes, both at the surface and the TOA. As a consequence, although time series of monthly bulk flux anomalies from the two NWP models and COADS agree very well over the northern extratropical oceans, these products were all found to contain large systematic biases that make them unsuitable for determining 16 net and the atmospheric transport. From Trenberth and Caron (21).

Wm -2 or K 32 CLIMATOLOGY from the ENERGY BALANCE MODEL 3 28 26 2 22 2 18 16 1 Insolation T e T g =.76 T g w/ transport and =.76 12-1 -5 5 1 LATITUDE 175 176 177 178 179 FIG.. The zonally and annual average insolation (blue curve) and solutions to the various energy balance models: (red) the emitting temperature (Eq. (5)); (yellow) with an absorbing atmosphere but no circulation (Eq. (1) with D = ); and (purple) with an absorbing atmosphere and circulation (D = Q o cos(3φ) in Eq. (11), where Q o = 5 Wm 2 and φ is latitude). The units are Wm 2 and K for insolation and temperature, respectively. 17

LATITUDE 8 7 6 5-2 - ANNUAL CYCLE SST (c.i., 1.K/.5K) 5 5 - -3 3-3 3-2 2-1 -.1 2 3 1-1 1.1 2 -.2 -.2.2 1 -.1.2 -.1.1 -.1 -.1.1.1.1.1 -.1.5 1 1.5 2 TIME (YEARS) 18 181 182 183 18 FIG. 5. Solution to the linearized time-dependent energy balance model Eq. (31), forced by the annual cycle of insolation (with the time mean removed). In this calculation, the mixed layer depth is taken to be 75m, α =.3, ε =.76, and the model has been linearized about the nonlinear solution to the equilibrium solution with idealized meridional energy transport is in Fig.. Contour interval is...,-2, -1, -.3, -.2, -.1,,.1,.2,.3, 1, 2,... C. 18

LATITUDE 8 ANNUAL CYCLE SST (c.i., 1.K/.5K).1.1 7 -.1 -.2.2.1 6 5-2 2-1 1 3-2 2 2 -.1 -.1-1 1 1.1.1 -.2 -.1 -.1.1.2 -.1.5 1 1.5 2 TIME (YEARS) 185 186 FIG. 6. As in Fig. 5, but including a simple parameterization of mixed layer depth as a function of latitude. 19

FIG. 7. Observed seasonal cycle zonally averaged SST. Contour interval is 2 C. 2