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A Simple Model for the Seasonal Cycle of Energy Fluxes in the high latitudes. Aaron Donohoe and David S. Battisti Thanks to Dargan Frierson and Arnaud Czaja. Photo: Ed WB IV. 5.7 PW. 5.9 PW. How do the absorbed solar (ASR), outgoing longwave (OLR), and meridional

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a simple model for the seasonal cycle of energy fluxes in the high latitudes
A Simple Model for the Seasonal Cycle of Energy Fluxes in the high latitudes

Aaron Donohoe and David S. Battisti

Thanks to Dargan Frierson and Arnaud Czaja

Photo: Ed WB IV

slide2

5.7 PW

5.9 PW

How do the absorbed solar (ASR), outgoing longwave (OLR), and meridional

heat transport (MHT) each contribute to the high latitude balance?

How does this picture change seasonally?

slide3

OUTLINE

I) Annual, zonal averaged energy flux (obs)

2) Zonal mean seasonal energy balance

Observed, 3-box EBM, aquaplanet GCM

3) Land-ocean contrasts and seasonal energy balances

Observed, 6-box EBM

4) Conclusions

slide4

1) Zonal and Annual Averaged Energy Flux (global mean removed)

ASR = Absorbed

solar

SHF = Surface

heat flux

(-)OLR = Outgoing

longwave

MHT = Meridional

Heat Transp.

CTEN = (-) Atmos

Column tendency

Observed NH

  • All signs defined wrt the atmosphere (e.g., positive OLR is an energy flux to the atmosphere)
  • Surface heat flux = the total energy flux (radiative plus turbulent) through the surface/atmosphere interface
  • In the annual mean, positive SHF is equal to oceanic heat flux convergence
annual mean of energy budget poleward of 30 0 n departures from global annual mean
Annual mean of energy budget poleward of 300N(Departures from global annual mean)

Absorbed

Solar (ASR)

2.2 PW

Surface Heat

Flux (SHF)

7.9 PW

Negative

OLR

4.3 PW

North Polar

Region

Tropics

Meridional

Heat Transport

(MHT)

1.4 PW

slide6

ASR

SHF

MHT

2) Zonal and Seasonal Averaged Energy Flux (zonal, annual average removed)

Observed NH

(-) OLR

CTEN

(-) Atmospheric Column

tendency

slide7

North Polar Region Seasonal Energy Balance

(Seasonal Anomalies from local annual mean)

JULY

JANUARY

15.8 PW

13.9 PW

3.0 PW

3.4 PW

1.8 PW

0.2 PW

2.3 PW

0.8 PW

8.4 PW

9.8 PW

ocean

ocean

(-) OLR

ASR

SHF

MHT

(-) CTEN

how do we understand the seasonal partitioning of energy flux on an equator to pole scale
How do we understand the seasonal partitioning of energy flux on an equator-to-pole scale?
  • What parameters & physics dictate the amount of energy (insolation) that goes into ocean storage vs. that that enters the atmosphere to drive temperature, OLR, and MHT changes?
  • What controls the ratio OLR:MHT:CTEN?
  • STRATEGY: build a ‘toy’ model; a linear 3-box model Energy Balance Model (EBM) with only the essential processes
an atmospheric layer in radiative equilibrium
An atmospheric layer in radiative* equilibrium

Space

Absorbed

atmosphere

LH =

BLH(TS-CLH)

Atmosphere

SENS =

BSENS(TS-TA-CSENS)

Surface

Solar Radiation

Terrestrial Radiation

Our EBM has three atmospheric layers - this is a schematic

the ebm basic state
The EBM basic state
  • Emissivity is a function of basic state temperature, assumed fixed relative humidity (75%) and CO2 according to Emmanuel (2002)

TA3= 225 K

TA2= 248 K

TA1= 262 K

TS= 287 K

Surface (black body)

the linearized basic state

4σ TS3TS’

The linearized basic state
  • We fix the layer emissivities but take the water vapor feedback into effect

ASR’ (prescribed)

3

TA’

CWV

4

CWV

3

4

TA’

LH’ = BLHT’S

SENS’ =

BSENS(TS’-TA’)

4 TS3TS’

4 TS3TS’

Sensible heat affects lowest atmosphere layer; latent distributed

vertical energy exchanges in the perturbed system
Vertical energy exchanges in the perturbed system

Change in OLR if the Column heats up uniformly

BOLR = BOLR,S +[BOLR,A]

= 2.6 Wm-2K-1

If only the surface heats up

0.8 Wm-2 K-1

1.8 Wm-2 K-1

BOLR,S

[BOLR,A]

BLW↑,S = 5.3 Wm-2 K-1

BLH = 4.0 Wm-2 K-1

BSENS = 3.0 Wm-2 K-1

linearized meridional heat transport

TROPICS

POLAR REGION

BMHT(TA3,T-TA3,P)/3

BMHT(TA2,T-TA2,P)/3

BMHT(TA1,T-TA1,P)/3

Linearized Meridional Heat Transport
  • Diffusive in each atmospheric layer:

MHT = BMHT ( [ T’A,Tropics] - [ T’A, Polar] )

(brackets = column mean)

  • BMHT = 3.4Wm-2K-1 (found by fitting obs)
motivation for diffusive heat transport
Motivation For Diffusive Heat Transport

Monthly atmospheric heat transport into polar region vs. polar – tropics atmospheric temperature difference

Similar results hold for pre-industrial climate, 4xPI CO2 and the LGM

annual mean polar region energy balance
Annual mean polar region energy balance
  • Assume no vertical temperature structure
  • Global mean energy balance requires:

OLR’ = 0 or TT = - TP = -ΔT

ASR’

ΔT

OLR’

-ΔT

MHT’

TROPICS

POLAR

So, MHT = BMHT2ΔT andOLR’ = BOLRΔT

Hence, MHT / OLR = 2BMHT/BOLR = 2.3

[Observed ratio is 2.6 (including ocean heat transport)]

what to compare the ebm to
What to compare the EBM to?
  • Real world is complicated by land-ocean contrast (later)
  • We compare our simple model runs to an aquaplanet slab ocean GCM
    • GFDL version 2.1
    • Ensemble with different slab ocean depths of 2.4, 6.0, 12.0, 24.0 and 50 meters
    • Average the output equatorward and poleward of 300
seasonal amplitudes vs slab ocean depth
Seasonal amplitudes vs. slab ocean depth

Asterisk =

Aquaplanet

GCM results

Solid Line =

EBM

Seasonal Amplitude of Temperature

Seasonal Amplitude of Energy Flux

These sum, in quadrature, to ASR

Slab Ocean Depth (m)

seasonal amplitudes vs slab ocean depth1
Seasonal amplitudes vs. slab ocean depth

Why is the seasonal amplitude of atmospheric temperature nearly equal to that of surface temperature?

What controls the quantity of energy that gets stored in the ocean vs. what gets fluxed to the atmosphere?

What controls the partitioning of the seasonal amplitude of MHT, OLR and CTEN?

Seasonal Amplitude of Temperature

Seasonal Amplitude of Energy Flux

Slab Ocean Depth (m)

slide21

i) Why is the seasonal amplitude of atmospheric temperature nearly equal to that of surface temperature?

Asterisk =

Aquaplanet

GCM results

Solid Line =

EBM

Step I : If we put energy into the atmosphere, negative feedbacks go to work:

As a result, the atmosphere come to equilibrium on a time scale much

shorter than the seasonal cycle

slide22

i) Why is the seasonal amplitude of atmospheric temperature nearly equal to that of surface temperature?

Step II : We can account for the (small) quantity of energy stored by the

atmosphere by:

Assume the temperature variations are annual,

then the tendency of energy associated with annual cycle of atmospheric temperature is:

slide23

LH

LW↑

SENS

CTEN

MHT

LW

SENS

i) Why is the seasonal amplitude of atmospheric temperature nearly equal to that of surface temperature?

Step III: The atmosphere is nearly in equilibrium at all times

Equating the input and output

  • The near equivalence in the seasonal amplitude of atmosphere and surface temperatures is a consequence of model parameters – our choices capture the GCM behavior
slide24
Implications

The component of the climate system that is the least efficient in exporting energy undergoes the largest seasonal temperature change

Example: What the atmosphere gets more efficient at exporting energy (ie, double BMHT)?

Our equation predicts:

repeat but reduce b mht by half
Repeat, but reduce BMHT by half

=0.85

The atmosphere must undergo a larger seasonal cycle to export a similar quantity of heat

revisiting b olr
Revisiting BOLR

The general linearized expression for OLR:

OLR’ = BOLR,S T’S+ [BOLR,A] [TA]’

OLR’ = BOLR [TA]’ , where BOLR == K BOLR,S + [BOLR,A]

and

[BOLR,A]

1.8 Wm-2 K-1

Note: 1/BOLR is the climate sensitivity when K =1.

[TA]’

BOLR,S

0.8 Wm-2 K-1

TS’

slide27
ii) What controls the quantity of energy that gets stored in the ocean vs. what gets fluxed to the atmosphere?

Concept:

  • The seasonal variations in absorbed solar radiation go directly into the ocean layer
  • Energy only gets fluxed to the atmosphere after the ocean surface heats up
  • The energy fluxed to the atmosphere is used to drive seasonal changes in MHT, OLR and CTEN
slide28

ii) What controls the quantity of energy that gets stored in the ocean vs. what gets fluxed to the atmosphere?

Step I: The seasonal amplitude of ocean energy fluctuations

= 33 W/(m2 K )

for a 40 meter deep ocean

So the seasonal amplitude of energy stored in ocean = BOClTSl

Step II : The energy fluxed to the atmosphere is the sum of the

seasonal variations in MHT, OLR, and CTEN

(BMHT+BOLR+BCTEN) lTAl

These add in quadrature to equal the seasonal amplitude of ASR

(BMHT+BOLR+BCTEN)2 lTAl2 + BOC2lTAl2κ2 = lASR’l2

slide29

What controls the quantity of energy that gets stored in the ocean vs. what gets fluxed to the atmosphere?

  • Given κ and B coefficients, we can figure out lTAl and lTSl
  • Therefore, we can determine the amount of seasonal solar
  • energy that is stored in the ocean vs. the seasonal energy
  • that is fluxed into the atmosphere

Dashed lines

are the result of

these equations

Solid lines = EBM

* = AGCM + slab

iii what controls the petitioning of the seasonal amplitude of mht olr and cten
iii) What controls the petitioning of the seasonal amplitude of MHT, OLR and CTEN?

Assuming tropical temperature doesn’t vary much seasonally, the seasonal cycle of atmospheric heat transport, OLR and atmospheric energy tendency will scale as

BMHT : BOLR : BCTEN = 3.4 : 2.6 : 2.0

at shallow slab ocean depths the seasonal amplitude exceeds the annual mean
At shallow slab ocean depths, the seasonal amplitude exceeds the annual mean

The annual mean MHT from the EBM is 5.5 PW

For ocean depths shallower than about 12 meters, our model predicts

that the equator to pole temperature gradient and heat transport changes

sign seasonally!

the pole to equator solar insolation gradient does change signs seasonally
The pole to equator solar insolation gradient does change signs seasonally

Could the temperature gradient and MHT also changes signs? Or is it simply a property of the simple EBM?

aquaplanet gcm simulation 6m slab ocean july
The surface temperature gradient does change signs in summer

The ‘mid-latitude jet is Easterly

Baroclinic eddies transport heat out of the northern polar region

Aquaplanet GCM simulation6m slab ocean; JULY

SURFACE TEMPERATURE

K

300 hPa Zonal Velocity

m/s

3 land ocean contrast and seasonal energy fluxes observations
3) Land Ocean Contrast and Seasonal Energy Fluxes - OBSERVATIONS

Zonal Anomaly over Land

ASR = Absorbed

solar

SHF = Surface

heat flux

(-)OLR = Outgoing

longwave

MHT = Meridional

heat transp.

CTEN = (-) Atmos

column tendency

ZHT = Zonal

heat transport

Zonal Anomaly over Ocean

nh january energy flux schematic observations
NH January energy flux schematic- OBSERVATIONS

Zonal averaged energy flux balance (annual mean removed)

LAND & OCEAN sub-domains

(zonal average removed)

LAND

OCEAN

13.9 PW

3.0 PW

0.4 PW

0.4 PW

0.3 PW

0.3 PW

0.1 PW

4.3 PW

0.1 PW

2.3 PW

0.2 PW

4.3 PW

4.3 PW

8.4 PW

ocean

WE ASSUME MHT IS ZONALLY INVARIANT

ASR

(-) OLR

CTEN

MHT

SHF

Zonal Heat Transport (ZHT)

model the observed climate system
Model the observed climate system
  • Specify land fractions of 0.5,0.25, and 0.10 in the NH polar region, tropics, and SH polar region
  • Ocean mixed layer depth fixed at 60 meters
  • Ocean and land both influence each other in the NH, Land can barely influence the ocean in the SH

N

FO,L= Land, Ocean Fraction

BZHT /FO,L= 20Wm-2 K-1 in NH

BMHT = 3.4 Wm-2 K-1

ZHT is “FAST”

how does the land ocean contrast influence the energy transport in the polar domains
How does the land/ocean contrast influence the energy transport in the polar domains?

Solid =

Observations

(ERBE/NCEP)

Dashed = 6-box

“control” EBM

All terms are

anomalies

from the global

annual mean

slide38

The energy Balance in the land and ocean subdomains

Solid = observations Dashed = EBM

All terms are local seasonal anomalies in W/m2

how do the energy fluxes and climatology respond to changes in land fraction in the polar region
How do the energy fluxes and climatology respond to changes in land fraction in the polar region?
  • Take an ensemble of EBM runs with different polar land fractions (equal in both hemispheres)

More seasonal variations

Of temp/OLR/MHT/CTEN

More Seasonal

Energy fluxed to atmos

More Land

slide40

Seasonal amplitude of surface and atmospheric temperature

  • Over Land: the seasonal amplitude of surface temperature exceeds that of atmospheric temperature
    • vice versa over the ocean

Polar Land Fraction

slide41

If we had isolated ocean and land domains

1.3

TA’ =

20K

TA’ =

5K

TS’ =

5K

TS’ =

26K

Ocean

Land

slide42

If we remove the barrier between the ocean and land

Κ goes up

Κ goes down

TA’ =

20K

15K

TA’ =

5K

10K

TS’ =

5K

6K

TS’ =

26K

24K

Ocean

Land

4 conclusions
4. Conclusions
  • In the annual mean, the polar region deficit in solar radiation is balanced by MHT and OLR in approximate ratio of 2.5 : 1
    • This ratio can be understood in terms of the relative BMHT and BOLR coefficients
  • The seasonal cycle of ASR in the polar region is primarily balanced by ocean heat storage; MHT, OLR, and CTEN play a decreasingly important role in the seasonal energy balance
    • The relative importance of each of these process can be understood in terms of the relative magnitudes of BOC, BMHT, BOLR, and BCTEN
4 conclusions cont
4. Conclusions (cont)
  • In the zonal mean, as the polar region land fraction increases, more seasonal energy is supplied to the atmosphere and the seasonal amplitude of temperature, MHT, and OLR increase
  • The small land surface heat capacity causes ASR to be balanced by ZHT to the ocean domain where the energy is transferred into the surface ocean
mid summer both hemispheres asr shf seasonal energy supplied to the atmosphere
Mid-summer (both hemispheres) ASR’ +SHF= Seasonal Energy supplied to the atmosphere

W/m2

  • If we stopped the atmospheric circulation – the atmosphere over
  • the high latitude ocean would cool in the middle of the summer
  • Energy that is transported by the atmosphere (meridionally and
  • zonally) goes (primarily) into seasonal storage in the polar ocean
  • This buffers the climate system’s seasonal cycle of temperature
slide47

Seasonal amplitude of surface and atmospheric temperature

  • Over Land: the seasonal amplitude of surface temperature exceeds that of atmospheric temperature
    • vice versa over the ocean
  • WHY?
    • More seasonal energy enters the atmosphere over land leading to a larger of seasonal amplitude of temperature there (compared to the ocean)
    • Aloft, energy is fluxed zonally away from the land and to the ocean in the warm season
      • Vice versa in the cold season
    • Thus, zonal transport reduces the seasonal cycle of air temperature over land and enhances it over ocean

Polar Land Fraction

add land and ocean subdomains to the 3 box ebm
Add land and ocean subdomains to the 3 box EBM

BZHT= 10 W/(m2 K)

  • The heat transport between the ocean and land domain is diffusive in each atmospheric layer:

Fo,L= Land or Ocean Fraction

Effective BZHT for NH is 20 W/(m2 K)

POLAR REGION

OCEAN

LAND

BZHT(TA3,O-TA3,L)/3

BZHT(TA2,O-TA2,L)/3

LH is turned

off over land

BZHT(TA1,O-TA1,L)/3

how to incorporate zonal heat transport into the ebm
How to incorporate zonal heat transport into the EBM
  • In the observed monthly data, the heat transport divergence associated with land/ocean advection (at each latitude) scales linearly with atmospheric temperature contrast between the land and ocean
  • The slope in the NH mid-latitude is 20 W/(m2 K)
  • We speculate that the offset between the fits at different latitudes is due to water vapor transport (Fasullo and Trenberth, 2008)
seasonal amplitude of surface and atmospheric temperature

LH

LW↑

SENS

CTEN

MHT

LW

SENS

Seasonal amplitude of surface and atmospheric temperature
  • Review: In aquaplanet mode, lTS’l/ l[TA’]l is dictated by: energy in = energy out

ENERGY INTO ATMOS

If we enhance the efficiency of

atmospheric heat export – the surface

temp. undergoes larger seasonal

Variations than the atmos/

ENERGY EXPORTED

BY ATMOS

DOUBLE BMHT Experiment

slide51

Seasonal amplitude of surface and atmospheric temperature

ENERGY IN

ENERGY OUT

LW↑

SENS

CTEN

MHT

ZHT

LW

SENS

If there were no seasonal cycle of

atmos temp. over the ocean,

atmos steady state over land says

LAND

Polar Land Fraction

This predicts KL = 4, which is too big

slide52

Seasonal amplitude of surface and atmospheric temperature

CTEN

MHT

LW

SENS

Closure of the problem requires knowledge

of the ratio:

Then, over the LAND:

> 1

And over the OCEAN:

< 1

OCEAN

ENERGY IN

ENERGY OUT

ZHT

LW↑

Polar Land Fraction

SENS

LH

slide54

Revisit the seasonal energy balance in the North Polar Ocean

ASR + SHF = energy put into

atmosphere

  • The seasonal amplitude of heat flux to the ocean exceeds the ASR seasonal cycle
  • In the absence of atmospheric heat transport, the atmosphere over the ocean would cool in the middle of the summer
  • In order for the ocean to warm the land in the winter, it must store some of the solar insolation reaching the land during the summer
annual mean of energy budget poleward of 30 0 n anomalies from annual global mean
Annual mean of energy budget poleward of 300N(Anomalies from annual global mean)

Absorbed

Solar (ASR)

7.9 PW

North Polar

Region

2.2 PW

Surface Heat

Flux (SHF)

4.3 PW

1.4 PW

Negative

OLR

Tropics

Meridional

Heat Transport

(MHT)

South

Polar Region

model the observed climate system1

N

ZHT

BZHT/FL([TO] - ([TL])

MHT

BMHT/FL([TT] - ([TP])

Model the observed climate system
  • Specify land fractions of 0.5,0.25, and 0.10 in the NH polar region, tropics, and SH polar region
  • Ocean and land both influence each other in the NH, Land can barely influence the ocean in the SH
the linearized basic state1

BOLR,STS’

ASR’ (prescribed)

BLW↑TA’

SENS’ =

BSENS(TS’-TA’)

3

LH’ = BLHTS

Atmosphere

4

3

4

BOLR,STS’

BLW↓TA’

4 TS3TS’

BOLR,STS’

The linearized basic state
  • We fix the layer emissivities but take the water vapor feedback into effect
motivation for diffusive heat transport1
Motivation For Diffusive Heat Transport

Monthly atmospheric heat transport into polar region vs. polar – tropics atmospheric temperature difference

revisiting b olr1
Revisiting BOLR

Change in OLR if the atmosphere

and surface heat independently

Change in OLR if the Column heats

up uniformly

BOLR = BOLR,S +[BOLR,A]

= 2.6 W/(m2 K)

BOLR = K BOLR,S + [BOLR,A]

= 2.6 W/(m2 K)

OLR’ = BOLR[TA]’

(in this case, 1/BOLR = climate sensitivity)

0.8 W/(m2 K)

1.8 W/(m2 K)

0.8 W/(m2 K)

1.8 W/(m2 K)

[BOLR,A]

BOLR,S

[BOLR,A]

BOLR,S

[TA]’

TS’=K TA’

slide62

If we remove the barrier between the ocean and land

Κ goes up

Κ goes down

TA’=20K

TA’=15K

TA’=5K

TA’=10K

TS’=5K

TS’=26K