1. Sources of CAM Arctic Surface Climate Bias Deduced from the Vorticity and Temperature Equations -- DJF
Richard Grotjahn and Linlin Pan
Dept. of LAWR, Univ. of California, Davis
In this talk we shall focus on new calculations since those shown in January 2008 AMWG and June 2007 CCSM workshop.
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2. Snapshot of CAM3 DJF Bias
q = ln(Psfc) bias
T42 bias (CAM3 AMIP minus ERA-40) shown for U, V, T. (NCEP RAI used for q)
Bias has prominent values in NE Atlantic.
Over N Europe: SLP & T too low, ζ too high
Over Barents Sea/NW Russia region: opposite
Over Beaufort Sea: SLP too low
Representative level shown for U, V, & T
U σ =0.5
V σ =0.5
T σ =0.5
Bias has multiple poles primarily in the Arctic and in the North Atlantic. The well-known model bias where the downstream end of the N Atlantic storm track enters N Europe instead of heading further north is seen in vorticity bias (not shown), SLP dipole straddling Scandinavia, and in the cold temperature bias of the N Atlantic.
The zonal wind bias will also be noted later.

3. Contributors to the Temperature bias equation:
T bias equation is CAM T equation minus ERA-40 T eqn.
= TC – TE . Where TC is CAM3, TE is era-40 DJF temperature
Term 1 has CAM bias and ERA-40 combinations found in a linear stationary wave (LSW) model such as Branstator (1990)
Term 2 (nonlinear bias terms)
Term 3 (all transient terms)
diabatic heating bias. QC & QE use independent eqn
T equation in a LSW model is Term 1 + forcing term
We formulate an equation for the temperature bias (equals T hat) by subtracting the T equation using CAM data from the corresponding T equation using ERA-40 data. This is then time averaged to get the top equation.
The terms are placed into four groups. The group labeled term1 are terms explicitly kept in a linear theoretical model to be discussed later. Term 2 has all nonlinear combinations of the bias terms. Term 3 has all transient heat flux terms from both CAM and ERA-40.
The diabatic heating bias is NOT calculated as a residual of this equation. It is calculated as a residual of a potential temperature equation shown on the bottom line. This follows the Hoskins et al 1989 approach. Q hat is thus somewhat independently estimated.
We next show which groups in the temperature equation are larger.

4. T bias eqn terms upper troposphere σ=0.3, DJF
LSW terms
Nonlinear terms
For upper troposphere along each midlatitude storm track:
LSW terms mainly
balanced by transient fluxes (lower right) &
diabatic heating (lower left)
Nonlinear terms NOT as important in T bias equation
The terms in the LSW are the larger terms in much of troposphere.
The nonlinear terms not kept in a LSW model are small N of 20N and cannot balance the LSW terms.
The transient terms are small in the lower troposphere but more important in the upper troposphere as shown here. The transient terms are larger in parts of the midlatitude storm tracks (Atlantic and to a lesser extent the Pacific).
The diabatic heating bias balances the LSW terms in the tropics. For areas N of 20N diabatic bias is important in the N Atlantic at this upper troposphere level. Diabatic bias is more important at mid tropospheric and lower tropospheric levels not shown on this slide.
Diabatic from θ eqn residual
Transient terms

5. T bias eqn terms mid- troposphere σ=0.5, DJF
LSW terms
Nonlinear terms
LSW terms large:
along midlatitude storm tracks
Arctic land areas
Nonlinear terms NOT important
LSW terms mainly balanced by:
transient bias (<0 then >0 along track)
diabatic bias (at Atl track end >0 reverse of σ=0.3 level)
NOTE THE SCALE CHANGES IN THE REMAINING FIGURES. THE EDGE IS AT 30N. PREVIOUSLY IT WAS THE EQUATOR.
At a middle troposphereic level the storm tracks are still important parts of the LSW model terms, though now transient bias terms are less important than above.
In Atlantic storm track:
The diabatic bias reversed sign on the downstream end from sig=0.3 level and is a notable contributor to the LSW pattern.
Diabatic bias still <0 on upstream end.
Nonlinear terms still negligible
Transient terms
Diabatic from θ eqn residual

6. T bias eqn terms near Earth surface σ=0.95, DJF
LSW terms
Nonlinear terms
LSW terms large:
along midlatitude storm tracks
Arctic land areas
Barents Sea
Nonlinear terms have import over Arctic lands
Transient bias:
Dipolar Pac coast
>0 Arctic lands
Diabatic bias:
< 0 N of 65N
>0 Pac, Scand.
At the lowest level tracked, all the terms have large contributions.
Not clear if these are valid or due to CAM and ECMWF model differences, such as topography.
With that caveat, the nonlinear and transient terms oppose each other over much of Arctic lands,
The Transient and diabatic terms also oppose each other over much of Pacific coastal region (water & land dipole)
Diabatic seems to have cooling bias over Arctic Ocean surface. Will see independent evidence for that later.
Diabatic from θ eqn residual
Transient terms

7. T-bias Eqn. Conclusions
In most of troposphere, LSW bias (term 1) is larger term.
Nonlinear (term 2) is comparable to term1 only near surface, mainly Arctic lands.
Transients (term 3) comparable to term 1 at upper troposphere, mainly storm tracks
Diabatic (term 4) large at troposphere and surface levels;
Transient & diabatic forcing key at surface & Atlantic storm track.
Conclusions to this point:
In most of troposphere, LSW bias (term 1) is larger term.
Nonlinear (term 2) is comparable to term1 only near surface, mainly Arctic lands.
Transients (term 3) comparable to term 1 at upper troposphere, mainly storm tracks
Diabatic (term 4) large at troposphere and surface levels;
Transient & diabatic forcing key at surface & Atlantic storm track.
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8. Contributors to the Vorticity bias equation:
Term 1 = Term Term Term 4
ζ bias equation is CAM ζ equation minus ERA-40 ζ eqn.
ζ^ = ζ C – ζE . Where ζ C is CAM3, ζ E is era-40 DJF temperature
Term 1 has CAM bias and ERA-40 combinations found in a linear stationary wave (LSW) model such as Branstator (1990)
Term 2 (nonlinear bias terms)
Term 3 (all transient terms)
Term 4 (diffusion/friction) F^ = FC – FE
ζ equation in LSW is Term 1 + forcing terms on RHS
We formulate an equation for the vorticity bias (equals zeta- hat) by subtracting the vort. equation using CAM data from the corresponding vort. equation using ERA-40 data. This is then time averaged to get the top equation.
The terms are again placed into four groups. The group labeled term1 are terms explicitly kept in a linear theoretical model to be discussed later. Term 2 has all nonlinear combinations of the bias terms. Term 3 has all transient vorticity flux terms from both CAM and ERA-40.
The diabatic bias is diffusion and friction and is calculated as a residual of this equation.
We next show which groups in the vorticity bias equation are larger.

9. ζ bias eqn terms mid- troposphere σ=0.5, DJF
LSW terms
Nonlinear terms
LSW pattern noisy:
Multiple poles N Atlantic & Greenland
>0 N side, <0 S side Atl. storm track
Nonlinear term large near: Iceland, Spain.
Transient bias <0 along tracks; Greenland dipole
Diffusion/Friction bias:
Reinforce nonlinear
Oppose transients
Representative middle & upper troposphere level.
The LSW model terms have a very noisy pattern. Somewhat expected for vorticity.
The nonlinear terms generally still not important, but there are now a few exceptions: Near Iceland and over Spain.
Transient terms tend force negative vorticity bias over the storm tracks. The negative values over GIN sea consistent with traveling frontal systems in CAM heading across N. Europe instead of further north.
Diabatic term has half the contour interval compared to other plots. Remember this is a residual. It is a noisy contributor to the LSW, amplifying the positive value in term 1 over Spain and the mess near Greenland.
Diabatic from ζ eqn residual
Transient terms

10. ζ bias eqn terms near Earth surface σ=0.95, DJF
LSW terms
Nonlinear terms
LSW terms dipoles:
Scand., Siberia, Bering St., Greenl’d
Nonlinear terms small
Transient bias small
Diffusion/Friction bias main balance to LSW:
Scand. dipole
Greenland poles
Bering strait dipole
At the lowest level tracked, all the friction/diffusion term dominates all other contributions to term 1.
Not clear if these are valid or due to CAM and ECMWF model differences, such as topography.
With that caveat, dipolar pattern over North Sea/Europe. This seems consistent with the track error: extracting the excess vorticity for simulated storms going into Europe, while less to remove over North Sea.
Diabatic from ζ eqn residual CONTOUR RANGE CHG.
Transient terms

11. ζ-bias Eqn. Conclusions
Linear (term 1) very noisy, dominates at upper levels and surface.
Nonlinear (term 2) important in spots
Transients (term 3) <0 along northern end of storm tracks (where CAM biases: U>0, ζ<0)
Friction term dominates at surface, downstream parts of storm tracks
Linear (term 1) very noisy, dominates at upper levels and surface.
Nonlinear (term 2) important in spots
Transients (term 3) <0 along storm tracks (where CAM biases: U>0, ζ<0)
Friction term dominates at surface, downstream parts of storm tracks
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12. Contributors to vertically-integrated diabatic bias:
T bias equation has large contribution from diabatic terms. But calculation as residual of θ equation is uncertain.
Trenberth & Smith (2008) recommend vertical integrals of the residual in T and moisture equations. Residual for T eqn is:
Calculate this for CAM & ERA-40.
Note that the vertical integrals also equal sums of boundary fluxes: TOA net radiation, surface sensible heat flux, L*Precipitation:
Compare bias calculated each way.
Bias from direct evaluation is:
The calculation of diabatic heating as residual of θ equation can be problematic. Finding the bias is therefore the difference between two residuals.
Trenberth & Smith (2008) recommend vertical integrals of the residual in T and moisture equations. The reason is one can compare a vertical integral of the equation residual with independent variables, namely top of atmosphere net radiation, surface sensible heat flux and the precipitation. Precip can be problematic, though Trenberth and Smith have a bit more confidence in P-E which enters from vertically-integrating a moisture equation.
We make calculations similar to theirs but for the model biases. A few results will be shown.
Some definitions: Q1^ comes from the T bias equation, while Q2^ is from a moisture equation, expressed as a heating rate. The heating rate in a total energy equation is their difference.
Note that the Q1^ definition can be partitioned into 3 parts, we show that next.
Compare individual parts of the Q1 bias.
Compare to ‘apparent’ heating of moisture eqn: Q2 = L*(Precip.– Evap.)
Compare to heating in Total energy eqn: Q1 – Q2

13. DJF vertically-integrated diabatic bias
Q1 bias dominated by heating bias (>0) over Atlantic, Pacific, W. coast (W/m2)
Net radiation bias (R) cooling (<0) over Arctic Ocean & Russia.
While >0 over Atlantic storm track mainly Precip., also Rad.
Pacific storm track <0 at start from SHF, >0 at end from P & R.
The T bias eqn diabatic heating is now vertically integrated.
The CAM precip is obviously too strong in the Atlantic storm track extending across Europe and Western Russia.
This is both a track orientation problem as well as excessive model precip problem.
CAM Precip also has a problem at the west coast of N. America, perhaps due to topographic differences in the model.
Sfc sensible heat flux bias generally a smaller term, except near start of storm tracks where it is different in the Pacific and Atlantic.
It is >0 over much of Russia
It is <0 off Japan suggesting less SHF in model than ERA-40
Net radiation bias is also >0 over Atlantic storm track, presumably the extra P has more cloudiness that absorbs more radiation (presumably at cloud bottom) and emits less (at cooler cloud top). Pattern in Pacific storm track somewhat similar.
However, net rad. is <0 over much of Mediterranean and southern former Soviet Union even though precip bias is >0 there.
Total of the terms is dominated by the P in the storm tracks. P and R reinforce over the storm track downstream end.
SHF and R appear to cancel over Arctic Ocean (surface energy budget difference?)

14. DJF vertically-integrated bias in ‘apparent’ heating & total energy
Q2 is heating deduced in a moisture eqn. Q2 is stronger in CAM over western Russia and part of N. Pacific.
Q1 – Q2 total energy eqn heating bias
N Atlantic track >0
Pacific track start, >0 downstream
Bias 
The apparent heating from moisture processes is >0 over Europe and western Russia, again reflecting the storm track error.
Compared with Q1 it may suggest excess precip is being offset by excess evaporation over Atlantic (but not over the downstream land area)

15. Vertically-integrated heating Conclusions
In T eqn heating bias, Q1: the part from precip (P) > net radiation (R) and both >0 over the N. Atlantic storm track.
R & P somewhat oppose over Europe and western former Soviet Union.
R <0 over much of Arctic Ocean, opposed by SHF
N. Atlantic storm track dominates bias in diabatic heating for T bias and total energy bias, (N. Pacific track different bias at start, not enough SHF in model)
In T eqn bias, the P > R and both >0 over the N. Atlantic storm track.
R & P somewhat oppose over eastern Russia.
R <0 over much of Arctic Ocean, opposed by SHF
N. Atlantic storm track more problematic, N. Pacific track somewhat different bias at start, (resolution problem at end with Rockies.)
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16. Overall Conclusions
Analyses of the T bias equation and the ζ bias eqn support using a LSW model to study the forcing responsible for the bias. (Though nonlinear terms a bit larger in ζ bias eqn. and near Earth’s surface)
Analysis of T & ζ bias equations:
Finds different behavior in Atlantic and Pacific storm tracks
Finds larger forcing of Arctic bias in Atlantic storm track,
especially in T eqn.
from diabatic and transient processes
Finds some forcing of Arctic bias locally
Diabatic bias has large contribution by precipitation differences, but also net radiation, especially on downstream end of storm tracks.
Most terms have large contributions near surface (unclear if is resolution issue)
Recommendations:
Need to fix >0 bias in: precipitation, net radiation, and heat flux in Atlantic storm track (including downstream over Europe)
Need to fix track error of the Atlantic storm track (creates dipolar forcing that results in single-pole bias)
Revisit low level diabatic processes over the Arctic?
Analyses of the T bias equation and the ζ bias eqn support using a LSW model to study the forcing responsible for the bias. (Though nonlinear terms a bit larger in ζ bias eqn. and near Earth’s surface)
Analysis of T & ζ bias equations:
Finds different behavior in Atlantic and Pacific storm tracks
Finds larger forcing of Arctic bias in Atlantic storm track,
especially in T eqn.
from diabatic and transient processes
Finds some forcing of Arctic bias locally
Diabatic bias has large contribution by precipitation differences, but also net radiation, especially on downstream end of storm tracks.
Most terms have large contributions near surface (unclear if is resolution issue)
Recommendations:
Need to fix >0 bias in: precipitation, net radiation, and heat flux in Atlantic storm track (including downstream over Europe)
Need to fix track error of the Atlantic storm track (creates dipolar forcing that results in single-pole bias)
Revisit low level diabatic processes over the Arctic?
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17. The End
Thanks for your patience!

18. Storage below
Slides below may be useful if there are questions.

19. Snapshot of CAM3 DJF Bias
q = ln(Psfc) bias
T42 bias (CAM3 AMIP minus ERA-40) shown for U, V, T. (NCEP RAI used for q)
Bias has prominent values in NE Atlantic.
Over N Europe: SLP & T too low, ζ too high
Over Barents Sea/NW Russia region: opposite
Over Beaufort Sea: SLP too low
Representative level shown for U, V, & T
U σ =0.5
V σ =0.5
T σ =0.5
Bias has multiple poles primarily in the Arctic and in the North Atlantic. The well-known model bias where the downstream end of the N Atlantic storm track enters N Europe instead of heading further north is seen in vorticity bias (not shown), SLP dipole straddling Scandinavia, and in the cold temperature bias of the N Atlantic.
The zonal wind bias will also be noted later.

20. DJF vertically-integrated diabatic Q1 & bias
Left panels: diabatic heating from vertical integral of the residual in T eqn.
Right panels: diabatic heating from direct measures of P, R, SHF.
Bottom is bias.
So essentially same bias for either calculation of Q1

21. DJF vertically-integrated diabatic Q1 & bias
Difference between calculating Q1 from vertical integral of the residual in T eqn. versus diabatic heating from direct measures of P, R, SHF.
So essentially same Q1 bias for either calculation

22. DJF vertically-integrated diabatic Q1 & bias
Left panels: diabatic heating from vertical integral of the residual in T eqn.
Right panels: diabatic heating from direct measures of P, R, SHF.
Bottom is bias.

23. Conclusions A temperature bias equation has been examined.
T bias eqn terms comprise 4 groups:
linear combinations of the T bias,
diabatic heating bias,
transient heat fluxes bias,
nonlinear (bias x bias) terms.
The diabatic heating and and transient eddy groups are prominent at latitudes higher than 30N (mainly in the midlatitude storm tracks) Nonlinear bias terms are small
The diabatic heating and transient heat fluxes are consistent with the well know storm track error in the N Atlantic.
The linear terms are included in a linear stationary wave (LSW) model, so such a model can be used to study the bias as if the bias were a stationary wave pattern. (The LSW model also includes divergence, vorticity, and log of surface pressure equations.)
LSW model severely limited by low resolution but previously we showed:
Bias in Arctic can be reproduced from forcing in Arctic and the N Atlantic.
Vorticity & T forcing have larger contribution than divergence and ln(Ps) forcing.
Future work may include testing the forcing in CAM from this T eqn analysis and from a corresponding vorticity eqn analysis.
A temperature bias equation was formed and examined. The T bias eqn terms comprise 4 groups:
1. linear combinations of the T bias and the analysis,
2. diabatic heating bias,
3. transient heat fluxes bias,
4. nonlinear (in the bias) terms
The diabatic heating and and transient eddy groups are prominent at latitudes higher than 30N (mainly in the midlatitude storm tracks) Nonlinear bias terms are small
The diabatic heating and transient heat fluxes are consistent with the well know storm track error in the N Atlantic.
The linear terms are included in a linear stationary wave (LSW) model, so such a model can be used to study the bias as if the bias were a stationary wave pattern. (The LSW model also includes divergence, vorticity, and log of surface pressure equations.)
LSW model is severely limited by low horizontal resolution but previously we showed:
The bias in the Arctic can be reproduced from using only the forcing in Arctic plus the forcing in the N Atlantic.
Vorticity & T forcing were found to have larger contribution to reproducing the input bias than did the divergence and ln(Ps) forcing.
Future work may include testing the forcing in CAM from this T eqn analysis and from a corresponding vorticity eqn analysis.