CAPE Singular vectors

Roel Stappers1 and Jan Barkmeijer

8th Adjoint workshop18 – 22 May 2009

Tannersville, PA, USA

1Stappers@knmi.nl1/23

MotivationOne of the main tasks of LAMs is prediction of high impactweather. Here we look at the possibility to develop a short rangeEPS in Hirlam which focuses on predictability of deep convection.

2/23

Overview

I TheoryI Singular vectorsI Convective Available Potential Energy (CAPE)I CAPE as final time norm

I Case studyI Experiment setupI Structure TE-SV versus CAPE-SVsI Test linearity assumption

I Conclusions and future plans

3/23

Singular vectorsLet x = g(x) and define ε(t) as

ε(t) = x(t)− x0(t)

Time evolution ε assumed linear

ε ≈ ∂g

∂x

∣∣∣∣x0(t)

ε

Integration from t = 0 to t = T

ε(T ) = M(0,T )ε(0)

Singular vectors are vectors ε(0)that maximize the ratio

||PMε(0)||C1

||ε(0)||C0

P is a projection operator.

This is equivalent to solving theeigenvector problem

M∗PC1PMv = σ2C0v

4/23

CAPE-norm

Let xr denote a reference modelstate and C a routine that computesCAPE

C(xr + ε)− C(xr ) ≈ ∂C∂x

∣∣∣∣xr

ε ≡ Cxr ε

The CAPE-norm is given by

||ε(T )|| =⟨Cx(T )ε(T ),Cx(T )ε(T )

⟩In the SV-calculation we also needthe adjoint of C

M∗G ∗C ∗x(T )Cx(T )GMε0 = σ2C0ε0

C and C ∗ are obtained using TAMC

5/23

ECMWF CAPE versus Hirlam CAPE

CAPE = g

∫θeup − θesat

θesatdz

30°N

30°N

40°N

50°N

60°N

70°N

80°N

80°W

80°W

60°W

40°W

20°W 0° 20°E

40°E

60°E

80°E

80°E

Tuesday 21 August 2007 18UTC ATHEN Forecast t+6 VT: Wednesday 22 August 2007 00UTC 999m

500

750

1000

1250

1500

1750

2000

2250

2398.9

ECMWF CAPE

CAPE = g

∫ EL

LFC

Tv (z ′)− Tv (z ′)

Tv (z ′)dz ′

30°N

30°N

40°N

50°N

60°N

70°N

80°N

80°W

80°W

60°W

40°W

20°W 0° 20°E

40°E

60°E

80°E

80°E

Tuesday 21 August 2007 18UTC ATHEN Forecast t+6 VT: Wednesday 22 August 2007 00UTC 50m

500

800

1200

1600

2000

2400

2800

3200

3459

CAPE 50 hPa

6/23

Aug. 22, 2007 6UTC: Forecast and measurements

FMI +6h forecast (1h Acc. Precip.) 5-8UTC Acc. Prec. Radar

The Finnish Hirlam model failed to forecast this thunderstorm inany cycle verifying at the same time (Pictures from T. Iversen)

7/23

KNMI Hirlam Conv. Prec. Aug 22, 6–12 UTC

12h forecast Conv. Precip.

16(AP18 − AP12)

6h forecast Conv. Precip.

16(AP12 − AP6)

8/23

Change in Max CAPE +018 - +012

9/23

SV Experiment settingsI Resolution: 0.5◦ × 0.5◦

I Optimization time: 12 hI Nonlinear trajectory updated every hour in TL-modelI Dry total energy norm at initial timeI Cape/TE-norm at final timeI Adjoint model uses Meteo France simplified physics:

Dry Moist

Vertical diffusion Vertical diffusionConvection

Large scale condensation

10/23

Leading TE Singular values dry TLM

11/23

Leading CAPE-Singular values dry TLM

12/23

Vertical energy distribution SVs Aug. 21 18 UTC

0 0.1 0.2 0.3 0.4 0.5

0

10

20

30

40

Initial SVs Total Energy norm

Energy

Mo

del

leve

l

0 1 2 3 4 5

0

10

20

30

40

Evolved SVs Total Energy norm

Energy

Mo

del

leve

l

0 0.02 0.04 0.06 0.08 0.1 0.12

0

10

20

30

40

Initial SVs CAPE norm

Energy

Mo

del

leve

l

0 0.5 1 1.5 2

0

10

20

30

40

Evolved SVs CAPE norm

Energy

Mo

del

leve

l

uvtq

Top(Initial SVs) Bottom(Evolved SVs) Left (TE) Right(CAPE)

13/23

Horizontal structure evolved SVs (10 SV average)

TE-SVs

30°N

40°N

50°N

60°N

70°N

80°N

80°W

80°W

70°W

60°W

50°W

40°W

30°W

20°W 10°W 0° 10°E 20°E

30°E

40°E

50°E

60°E

70°E

80°E

80°E

Vert. integrated kinetic energy

CAPE-SVs

30°N

40°N

50°N

60°N

70°N

80°N

80°W

80°W

70°W

60°W

50°W

40°W

30°W

20°W 10°W 0° 10°E 20°E

30°E

40°E

50°E

60°E

70°E

80°E

80°E

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

Vert. integrated specific humidity

14/23

Temperature evolved CAPE-SVs

30OE 32OE 34OE 36OE 38OE 40OE 42OE 44OE 46OE 48OE 50OE

58.0N

40

36

32

28

24

20

16

12

8

4

Cross section of temp 20070821 1200 step 0

60°N60°N

40°E

40°E

Tuesday 21 August 2007 12UTC ATHEN Analysis t+ VT: 12UTC Model Level 38 temperature

15/23

Twin experiments

Let

ε+(t) = x+(t)− x0(t)

and

ε−(t) = x−(t)− x0(t)

In a linear model we have

〈ε+(t), ε−(t)〉||ε+(t)|| ||ε−(t)||

= −1

Settings

I IC : ECMWF analysis

I No DFI no NMI

I Pert.: leading CAPE-SV

I Physics: Savijarvi, Straco,ISBA,CBR

I Scaling Tmax = 0.4 K, umax,vmax 0.3 m/s

16/23

Twin experiment specific humidity (t=12h)

-0.0

002

50°N

60°N

40°E

40°E 60°E

60°E

0.0001 (0.0004)

50°N

60°N

40°E

40°E 60°E

60°E

0.0001 (0.0004)

20OE 24OE 28OE 32OE 36OE 40OE 44OE 48OE40

30

20

10

-0.0

002

0.0002

0.00

02

0.00

04

Cross section of spec hum 20070821 1200 step 12

0.0001 (0.0004)

20OE 24OE 28OE 32OE 36OE 40OE 44OE 48OE40

30

20

10

0.00

02

0.0004

Cross section of spec hum 20070821 1200 step 12

0.0001 (0.0004)

17/23

Twin experiment temperature (t=12h)

30°N

40°N

50°N

60°N

70°N

80°N

80°W

80°W

60°W

40°W

20°W 0° 20°E

40°E

60°E

80°E

80°E

0.1 (0.4)

30°N

40°N

50°N

60°N

70°N

80°N

80°W

80°W

60°W

40°W

20°W 0° 20°E

40°E

60°E

80°E

80°E

0.1 (0.4)

20OE 24OE 28OE 32OE 36OE 40OE 44OE 48OE40

30

20

10

-0.2

0.2

Cross section of temp 20070821 1200 step 12

0.1 (0.4)

20OE 24OE 28OE 32OE 36OE 40OE 44OE 48OE40

30

20

10

0.2

Cross section of temp 20070821 1200 step 12

0.1 (0.4)

18/23

Twin experiment Temperature (t=0)

30°N

40°N

50°N

60°N

70°N

80°N

80°W

80°W

60°W

40°W

20°W 0° 20°E

40°E

60°E

80°E

80°E

0.01 (0.04)

30°N

40°N

50°N

60°N

70°N

80°N

80°W

80°W

60°W

40°W

20°W 0° 20°E

40°E

60°E

80°E

80°E

0.01 (0.04)

20OW 10OW 0O 10OE 20OE 30OE 40OE 50OE40

30

20

10

-0.02

Cross section of temp 20070821 1200 step 0

0.01 (0.04)

20OW 10OW 0O 10OE 20OE 30OE 40OE 50OE40

30

20

10

-0.02

Cross section of temp 20070821 1200 step 0

0.01 (0.04)

19/23

Twin experiment u-comp wind (t=1h)

30°N

40°N

50°N

60°N

70°N

80°N

80°W

80°W

60°W

40°W

20°W 0° 20°E

40°E

60°E

80°E

80°E

0.01 (0.04)

30°N

40°N

50°N

60°N

70°N

80°N

80°W

80°W

60°W

40°W

20°W 0° 20°E

40°E

60°E

80°E

80°E

0.01 (0.04)

20OW 10OW 0O 10OE 20OE 30OE 40OE 50OE40

30

20

10

-0.02

Cross section of u-comp 20070821 1200 step 1

0.01 (0.04)

20OW 10OW 0O 10OE 20OE 30OE 40OE 50OE40

30

20

10

-0.02

Cross section of u-comp 20070821 1200 step 1

0.01 (0.04)

20/23

Twin experiment Temperature (t=1h)

30°N

40°N

50°N

60°N

70°N

80°N

80°W

80°W

60°W

40°W

20°W 0° 20°E

40°E

60°E

80°E

80°E

0.01 (0.04)

30°N

40°N

50°N

60°N

70°N

80°N

80°W

80°W

60°W

40°W

20°W 0° 20°E

40°E

60°E

80°E

80°E

0.01 (0.04)

20OW 10OW 0O 10OE 20OE 30OE 40OE 50OE40

30

20

10

-0.1

4

-0.02

Cross section of temp 20070821 1200 step 1

0.01 (0.04)

20OW 10OW 0O 10OE 20OE 30OE 40OE 50OE40

30

20

10

-0.1

4

-0.02

-0.02

Cross section of temp 20070821 1200 step 1

0.01 (0.04)

21/23

Conclusions

I Both CAPE and TE SVs indicate that the 12 hour forecastsaround August 22 are sensitive to initial conditions.

I The TE-SVs in Hirlam show well known features: most energyin the temperature field at initial time and most energy in thewind-field at final time near the jetstream.

I CAPE-SVs are situated much lower in the troposphere and atfinal time the “energy” is mostly in the specific humidity field.12 hour CAPE forecast are most sensitive to the analysistemperature at 850 hPa.

I The twin experiments show that the tangent linearapproximation using CAPE-SVs as IC perturbations is valid upto 12 hours at 0.5◦ resolution

I The twin experiments show that there is “noise” in the entireboundary layer (expect for regions close to the lateralboundary) in the forecast after 1h

22/23

Future plans

I Further investigate the Finnish case (Higher resolution,OT=24h, etc)

I Investigate the noise in the twin experiment

I Modify CAPE-norm to include CIN

I Look at integrated water vapor as final time norm

I Further test linearity compare TE-SVs versus CAPE-SVs

I How to use (CAPE)-SVs as building blocks for EPS members(in GLAMEPS)?

23/23