A Simple Energy Balance Model of Climate

A Simple Energy Balance Model of Climate 1. The Basic zero-dimensional Energy Balance Model (EBM) for global average temperature Energy Input (sun) Energy Output (planetary radiation) Equilibrium: the Greenhouse Effect, climate Circulation 2. The one-dimensional time dependent EBM Response time scale Seasonal cycle, land and ocean 3. When does circulation matter? Temperature Precipitation Part II: Circulation

A Simple Energy Balance Model of Climate 1. The Basic zero-dimensional Energy Balance Model (EBM) for global average temperature Energy Input (sun) Energy Output (planetary radiation) Equilibrium: the Greenhouse Effect, climate Circulation 2. The one-dimensional time dependent EBM Response time scale Seasonal cycle, land and ocean 3. When does circulation matter? Temperature Precipitation

The Basic zero-dimensional Energy Balance Model (EBM) for global average temperature Solar Absorbed The amount of energy absorbed depends on the top of the atmosphere radiation S o = 1367 W m "2 the distribution of insolation incident on the spherical earth "(x,t) the optical properties of the absorbers (mainly in the visible bands, where most of the energy from the sun is emitted). Most of the objects that absorb visible (insolation) well are on the surface of the planet (only a small fraction of insolation is absorbed in the atmosphere)

The Basic zero-dimensional Energy Balance Model (EBM) for global average temperature Solar Absorbed The amount of energy absorbed depends on albedo " We will define albedo as the ratio of visible energy reflected (not used) to that incident will be called the albedo The average for the earth is called the planetary albedo Annual Average Albedo " earth = 0.30

Surface Albedo (snow-free)

The Basic zero-dimensional Energy Balance Model (EBM) for global average temperature Solar Absorbed The amount of energy absorbed at the surface of the earth is thus S o "(x,t) (1#$) If we are interested in the annual average temperature of the earth, the amount of incident radiation is just the shadow area of the earth " = # r 2 Hence and the total absorbed solar insolation is S o (1"#) $ r 2

The zero-dimensional EBM Emitted (Longwave) Radiation Though the atmosphere is a lousy absorber of visible radiation (i.e., insolation), it is a good absorber of infrared (Longwave) radiation that is emitted from the ground. Approximate the absorbing atmosphere as a slab with temperature T a that is emitting energy upward (to space) and downward (to the ground). The energy emitted is " # T a 4 where is a bulk emissivity and the area doing the emitting is 4" r 2.

The zero-dimensional EBM Equilibrium The energy balance at the surface is S o (1"#) ($ r 2 ) + % & T a 4 (4$ r 2 ) = % g & T g 4 (4$ r 2 ) or S o (1"#) /4 + $ % T a 4 = % T g 4 (1) where we have used " g =1 (a very good approximation). The energy balance in the atmosphere is 2 " # T a 4 = " # T g 4 + D (2) where we have included the term D to take into account heat flux convergence by the atmospheric circulation

The zero-dimensional EBM Equilibrium (without circulation) The solution without circulation is T g 4 = S o (1"#) 4$ (1"% /2) (3) 2 T a 4 = T g 4 (4) With no absorbing atmosphere (" = 0), the ground temperature is 255K. The correct answer, 288K, is obtained with " = 0.76. " = 0.30 ; S o =1367 Wm #2 ; $ = 5.67 x 10 #8 Wm #2 K #4

Annual Radiation Balance Differential local imbalances give rise to circulation

Meridional Energy Transport Ocean does most of the transport in the deep tropics (15ºS-15ºN) Atmosphere does most of the transport poleward of 30ºN,S

The one-dimensional EBM with circulation Equilibrium (with circulation) Since we are now interested in a local temperature, we need to revisit the surface energy budget (1): S (1"#) + $ % T a 4 = % T g 4 (5) where S is now the insolation incident at a particular latitude, and T a, T g are now a function of latitude The solution can be written " (2 #$ ) T a 4 = S (1#%) + D/$ (6) T g 4 = 2 T a 4 " D/(2#$) (7)

The one-dimensional EBM with circulation Circulation (ocean) cools the deep tropics and (atmosphere) warms the mid and high latitudes (Idealized) How much difference? About 10C in annual mean temperature

The one-dimensional EBM The Time Dependent One-Dimensional Equations Lets recall the latitudinal dependent energy budget equations, only now allow for time dependence C a " T a " t = # $ T g 4 % 2 # $ T a 4 (6) C g " T g " t = S (1#$) + % & T a 4 # & T g 4 (7) where S is the insolation incident at a particular latitude, C x is the heat capacity, and we are ignoring circulation The heat capacity of the atmosphere and ground (ocean) are C a = C p a P o /g = (10 3 J/kg K -1 ) (10 4 kg/m 2 ) = 10 7 J/m 2 K -1 C g = C ocean = C p o " water h = (4 x10 3 J/kg K -1 ) (10 3 kg/m 3 )(75m) = 30 x 10 7 J/m 2 K -1

The one-dimensional EBM The Time Dependent One-Dimensional Equations We can linearize the equations (6) and (7) by writing T g = T r + T g ' T a = T r + T a ' where T r is a constant (reference temperature): T r = 273.15K Dropping the primes, we have C a " T a " t = # a$ + b $ T g # 2 b $ T a (8) where C g " T g " t = S (1#%) # a (1#$) + b $ T a # b T g a = " T r 4, b = 4a/T r (9)

The one-dimensional EBM The Time Dependent One-Dimensional Equations Since C a << C o, over the oceans the atmosphere is essentially in equilibrium with respect to the ocean and we can simplify the atmospheric equation(8) as 0 = " a# + b # T g " 2 b # T a (8 ) $ T C g g = S (1"%) " a (1"#) + b # T a " b T g $ t and the equations simplify to (9) C o " T g " t = S (1#$) # A # B T g (10) where A = a (1"# /2) = 195 Wm "2 and B = b (1"# /2) = 2.9 Wm "2 C "1

The one-dimensional EBM The adjustment time scale Lets re-write equation (10) as follows F = S (1"#) " A = C o $ T g $ t + B T g (10) The general solution is where T g = e "t F # $ o e t # dt 0 t C o (11) " = C o B (12)

The one-dimensional EBM The adjustment time scale Consider the response of our system to a sudden forcing F given by The solution is F = F o t > 0 0 t " 0 (13) T g (t) = F o $ B 1" # ' %& e"t () " Hence, can be considered an adjustment time scale: " ocean = 3.3 years ; " land = 10 days ;

The one-dimensional EBM The seasonal cycle (without circulation) After removing the time-independent part of our linear equation (10 ), we have S (1"#) = C o $ T g $ t + B T g (14) Writing the annual cycle of insolation S as $ ( S, T ) g = Re ( S ˆ j, T ˆ ' &" ) gj exp (i # j t) ) %& j () the fourier solution to (14) is obtained: (15) C o i " j ˆ T gj = ˆ S j (1#$) # B ˆ T gj (16)

The one-dimensional EBM The seasonal cycle (without circulation) The solution in real space is where T g = Re[ T ˆ gj exp (i " j t) ] = S j (1#$) ( B 2 + (C o " j ) 2 ) 1 2 % C " j = tan #1 o $ j ( ' & B * ) cos (" j t # % j ) (17) (18) For the ocean, " j # $ /2 so temperature lags insolation by ~ 3 months. For the land, " j #10, so temperature lags insolation by ~10 days.

The one-dimensional EBM The seasonal cycle (without circulation) Using the simple energy balance model (without circulation) we would expect The annual cycle over the land to have a range of 72ºC and lag insolation by 10 days Use the simple one-dimensional EBM with observed annual harmonic in insolation and idealized mixed layer depth Using the simple energy balance model (without circulation) we would expect the annual cycle over the ocean to have a range of 4ºC in midlatitudes and lag the insolation by 3 months

Seasonal Cycle in Temperature 1-D EBM with variable mixed layer

Annual Cycle of observed SST

When does circulation matter for surface temperature? Experiment with an Atmospheric GCM coupled to a slab ocean w/ prescribed ocean heat flux convergence Compared to the zonal mean, Oceans are 20ºC colder than land NE Canada is 25ºC colder than over Europe Winds, SLP and Eddy Temperature in January Seager et al 2000

When does circulation matter for surface temperature? Experiments with an Atmospheric GCM coupled to a slab ocean w/ prescribed ocean heat flux convergence Forward Rotation Reverse Rotation NE Canada 25ºC colder than Europe NE Canada 5-10ºC warmer than Europe + 20ºC - 21ºC Reverse minus Forward

When does circulation matter for precipitation? Always, and everywhere z Precip Evap " 3

Conclusions: Part 1 1. Much of the Earth s climatological temperature is explained by local (radiative) processes Radiation absorbed = radiation emitted 2. Meridional heating gradients result from (relatively small) imbalances in absorved and emitted radiation Drives the atmosphere circulation (winds) that transport momentum, moisture and energy towards poles, cooling the tropics and warming the extratropics by >10ºC from what might be expected from radiative balance. Winds give rise to ocean circulation (part 2) that cools the tropics (by ~5 to 10ºC) below a the radiative balance.

Conclusions: Part 1 3. The seasonal cycle in temperature In the Southern Hemisphere, is what is expected from local processes (no circulation In the Northern Hemisphere, circulation matters Warms the ocean and cools the land in summer (cools the land and warms the ocean in winter) partly due to zonal winds (over land/ocean with differential heat capacity) partly due to standing waves (generated by orography) 4. The seasonal cycle in precipitation Circulation is key everywhere Part II: Circulation