Perturbation growth characterization in a convection-resolving system

workshop- Meteorological Institute of Belgium - Thursday 25 November 2010 Instability properties of regional models: ensemble forecast, boundary forcing and assimilation

Francesco Uboldi - Novate Milanese, MI, Italy; e-mail: uboldi@magritte.it

Anna Trevisan – ISAC-CNR, Bologna, Italy

Collaboration with the MOLOCH development team of ISAC-CNR, Bologna: Oxana Drofa, Silvio Davolio, Piero Malguzzi, Andrea Buzzi

➔ The convection-resolving non-hydrostatic model MOLOCH

➔ Breeding perturbations on a limited-area, non-hydrostatic system

➔ Perturbation growth characterization: growth exponent, doubling time;

time interval of validity of the tangent linear approximation

➔ Preliminary results of data assimilation experiments in idealized conditions

Perturbation growth characterization in a convection resolving system

➔ The convection-resolving non-hydrostatic model MOLOCH

➔ Breeding perturbations on a limited-area, non-hydrostatic system

➔ Perturbation growth characterization: growth exponent, doubling time;

time interval of validity of the tangent linear approximation

➔ Preliminary results of data assimilation experiments in idealized conditions

Perturbation growth characterization in a convection resolving system

MOLOCH is a non-hydrostatic, convection-resolving model developed by the atmospheric dynamics group of ISAC-CNR (Bologna).

Resolution about 2.3 km over northern Italy, including an Alpine area and portions of both the Ligurean Sea and the Adriatic Sea.

The control trajectory is a simulation of the real case 26 September 2007, with initial and boundary conditions from GFS.

The circulation at 500 hPa is characterized by a deep trough approaching the western Alpine region, and favouring the development of a shallow orographic cyclone centred over the Gulf of Genoa. A low level south-easterly jet flows over the Adriatic Sea. The Venice area is affected by intense convective precipitation, mostly during the morning: different scattered convective cells at the beginning, a well organized Mesoscale Convective System later.

Figure: total precipitation, accumulated from 00 to 12 UTC in the control run.

MOLOCH: Malguzzi et al., JGR-Atmospheres, 2006; Davolio et al., MAP, 2007; 2009; http://www.isac.cnr.it/dinamica/projects/forecasts/

State variables:T, p, q, u, v, w, cw, ci, pw, pi1, pi2 + ground T,q

➔ The convection-resolving non-hydrostatic model MOLOCH

➔ Breeding perturbations on a limited-area, non-hydrostatic system

➔ Perturbation growth characterization: growth exponent, doubling time;

time interval of validity of the tangent linear approximation

➔ Preliminary results of data assimilation experiments in idealized conditions

Perturbation growth characterization in a convection resolving system

Breeding

Before each renormalization: perturbation= perturbed state – control state

small initial perturbation;

nonlinear integration of the control

state and of the perturbed state;

periodic renormalization of the

growing perturbation to impose linear

growth (along the non-linear trajectory)

After each renormalization:perturbed state= control st. + renormalized pert.

Toth, Kalnay 1993

Initially independent perturbations collapse onto one direction (in an ideally infinite time: in fact, in a finite time, but it may be long)

PRACTICALLY IMPORTANT: MANY initially independent perturbations collapse onto FEW directions in a comparatively SHORT time!

A bred vector progressively assumes the structure of a linear combination of the most unstable directions

Initial coefficients (of the linear combination) unknown

(orthogonalization)

Breeding

Simplified scheme:1 two scales only2 growth early stages not described

Breeding

Nonlinear Breeding:filtering small and fast scales

Simplified scheme:1 two scales only2 growth early stages not described

Breeding

A perturbation breeding technique is used to estimate the local unstable structures

that characterize the dynamic growth of small perturbations and errors.

Breeding perturbations in a convection-resolving, limited area model:

Limited area models are forced by the lateral boundary conditions: the boundary

forcing tends to stabilize the trajectory;

Strong nonlinearities (non-linearizable processes) in moist thermodynamics

associated to convection

Maps: perturbation horizontal velocity module at model level 5 u2 v2

Two initial small, random, independent perturbations. Each variable is scaled with its variability.Rescaling perturbation amplitude is 0.05 m s-1 for horizontal velocity at model level 5(about 925 hPa over sea). Rescaling frequency is every 5 minutes.

After breeding for 1h30, the organized bred vectors show similar spatial structures, localized in dynamically active areas (intense winds and convective precipitation)➔ Bred vectors quickly get organized in spatially coherent structures.➔ The growth of small perturbations in the linear regime is not immediately disrupted by strongly non-linear processes present in moist convection. ➔ Similar results for different renormalization amplitudes and frequencies

➔ The convection-resolving non-hydrostatic model MOLOCH

➔ Breeding perturbations on a limited-area, non-hydrostatic system

➔ Perturbation growth characterization: growth exponent, doubling time;

time interval of validity of the tangent linear approximation

➔ Preliminary results of data assimilation experiments in idealized conditions

Perturbation growth characterization in a convection resolving system

eT d=2

NON-linear evolution of states perturbed by organized bred vectors

The growth of perturbations is characterized by estimating the growth exponent λ and the doubling time T

d :

The logarithm of the amplification factor after n time steps is approximately linear in

time during a period of linear growth regime.

Free non-linear evolution of two "opposed" perturbations (same direction, different

sign) after a breeding period (1h30').

At initial time:

At later times:

lnn≡ln∥ n∥∥0∥

=n n t

x p1 ,2=xc±

1,2=x p1 ,2−xc

T d=ln 2

RED, GREEN amplification factors for the two perturbations π1 and π

2.

BLUE CURVE...later

lnn≡ln∥ n∥∥0∥

=n n t

Td 1st exp:

Estimate of doubling time Td : T d=

ln 2⟨lnn⟩2h r. m.

n t

T d≃2.5 h

RED, GREEN amplification factors for the two perturbations π1 and π

2.

FIRST EXPERIMENT starts at 01h30.SECOND EXPERMENT starts at 06h00.

BLUE CURVES...later

lnn≡ln∥ n∥∥0∥

=n n t

Td 2nd exp:

Estimate of doubling time Td : T d=

ln 2⟨lnn⟩2h r. m.

n t

T d≃2 h

Tlin

: time period of validity of the tangent linear approximation along the non-

linear evolution

If Tlin

<< Td , small scale errors quickly saturate, when they are still small: in

these conditions, a reduction of the initial error does not correspond to a forecast improvement.

Linearity indicators (Hohenegger et al., BAMS 2006) are used to estimate .

Spatial correlation:

where:

The spatial correlation is computed for each variable separately.The scalar product is extended: A) to the whole domain to obtain global indicators;B) locally, in the neighbourhood (10 km) of each gridpoint.

corr 1 , 2=⟨1∣ 2⟩

∥ 1∥ ∥ 2∥

∥∥=⟨∣ ⟩ ⟨1∣ 2⟩=∑i

i1i

2

BLUE: spatial correlation

FIRST experiment: linearity is lost at

04h00, 2.5h after the initial time:

SECOND experiment: the spatial

correlation remains below the threshold

until the end of the simulation. Even

considering a stabilizing effect of the

boundary forcing after 10h00, the

estimate is still large:

T lin≃2.5 h

T lin4.0 h

FIRST experiment:

SECOND experiment:

The second convective episode, yet more intense, appears more predictable,

probably because large scale forcing has a more important role.

T lin≃T d≃2.5h

T lin2T d≃4.0 h

LEFT: perturbation 1 (wind at level 5) RIGHT: local spatial correlationsBoth fields are masked for small values of the perturbationAt 01h30 the local correlation is -1 everywhere.

the local correlation field shows loss of linearity (orange and red shading) over areas that do not correspond to the maximum growth regions (yellow shading in the left panel), but to moist thermodynamic processes and saturation of small scale growth.

➔ Strongly non-linear(izable) processes (moist thermodynamics, phase

transitions during convection) seem to affect predictability when they

succeed in determining non-linearity in the evolution of the horizontal

velocity field. Other variables (temperature, pressure, vertical velocity,

humidity and concentrations of condensed phases) are more directly

affected.

➔ Estimated values of Tlin

and Td also lead to some optimism, at least for

very short prediction range. Estimates, though, are dependent on the

amplitude of the initial perturbation. A larger perturbation amplitude

requires a larger domain to avoid (delay) the stabilizing effect of boundary

forcing.

➔ The convection-resolving non-hydrostatic model MOLOCH

➔ Breeding perturbations on a limited-area, non-hydrostatic system

➔ Perturbation growth characterization: growth exponent, doubling time;

time interval of validity of the tangent linear approximation

➔ Preliminary results of data assimilation experiments in idealized conditions

Perturbation growth characterization in a convection resolving system

data assimilation experiments in idealized conditions

Twin experiment:

A MOLOCH trajectory is chosen as "truth".

The "control" trajectory starts from a different initial condition (obtained by initializing a MOLOCH run at 2100 of the previous day).

SAME BOUNDARY CONDITIONS: no error from the boundary, only the evolution of initial error is considered.

PERFECT OBSERVATIONS are generated by applying an appropriate observation operator to the true state.

Moreover, observations are available where needed.

PURPOSE: assess how much (how possible) it is to control the growth of unstable structures present in the initial error, by assimilating observations.

Accumulated precipitation from 0000 to 1200 UTC in the true trajectory

Error field (difference "control" – "truth") at 0300 UTC. Color shading: wind difference module at level 5 (~925hPa over sea).Arrows: true state wind field.

Error field (difference "control" – "truth") at 0300 UTC. Color shading: wind difference module at level 8 (~850hPa over sea).Arrows: true state wind field.

Error field (difference "control" – "truth") at 0300 UTC. Color shading: wind difference module at level 15 (~700hPa over sea).Arrows: true state wind field.

Error field (difference "control" – "truth") at 0300 UTC. Color shading: wind difference module at level 25 (~500hPa over sea).Arrows: true state wind field.

AUS : Assimilation in the Unstable Subspace

Estimated unstable directions are available as N vectors en :

Impose that the analysis increment:

is confined in the estimated unstable subspace: x=Ea

E=[ e1 e1 e N ]

xa=x f

x

AUS : Assimilation in the Unstable Subspace

Estimated unstable directions are available as N vectors en :

Impose that the analysis increment:

is confined in the estimated unstable subspace:

minimize:

x=Ea

E=[ e1 e1 e N ]

xa=x f

x

2J=aT−1 a[H x f Ea−d ]T R−1[H x f Ea−d ]

AUS : Assimilation in the Unstable Subspace

Estimated unstable directions are available as N vectors en :

Impose that the analysis increment:

is confined in the estimated unstable subspace:

minimize:

The analysis is:

x=Ea

E=[ e1 e1 e N ]

xa=x f

x

2J=aT−1 a[H x f Ea−d ]T R−1[H x f Ea−d ]

xa=x f

E H ET[ H E H E

TR ]

−1d

AUS : Assimilation in the Unstable Subspace

Estimated unstable directions are available as N vectors en :

Impose that the analysis increment:

is confined in the estimated unstable subspace:

minimize:

The analysis is:

equivalent expression:

x=Ea

E=[ e1 e1 e N ]

xa=x f

x

2J=aT−1 a[H x f Ea−d ]T R−1[H x f Ea−d ]

xa=x f

E[ −1H E

T R−1H E ]

−1HE

T R−1d

xa=x f

E H ET[ H E H E

TR ]

−1d

AUS : Assimilation in the Unstable Subspace

Estimated unstable directions are available as N vectors en :

Impose that the analysis increment:

is confined in the estimated unstable subspace:

minimize:

The analysis is:

equivalent expression:

Case N=1

x=Ea

E=[ e1 e1 e N ]

xa=x f

x

2J=aT−1 a[H x f Ea−d ]T R−1[H x f Ea−d ]

xa=x f

eHe T R−1d

−1He T R−1

He

xa=x f

E[ −1H E

T R−1H E ]

−1HE

T R−1d

xa=x f

E H ET[ H E H E

TR ]

−1d

AUS : Assimilation in the Unstable Subspace

Estimated unstable directions are available as N vectors en :

Impose that the analysis increment:

is confined in the estimated unstable subspace:

minimize:

The analysis is:

equivalent expression:

Case N=1

Case

x=Ea

E=[ e1 e1 e N ]

xa=x f

x

2J=aT−1 a[H x f Ea−d ]T R−1[H x f Ea−d ]

xa=x f

eHe T R−1d

−1He T R−1

He

R=o2 I xa

=x fe

He T d

2He T He

2=o

2

xa=x f

E[ −1H E

T R−1H E ]

−1HE

T R−1d

xa=x f

E H ET[ H E H E

TR ]

−1d

AUS : Assimilation in the Unstable Subspace

Estimated unstable directions are available as N vectors en :

Impose that the analysis increment:

is confined in the estimated unstable subspace:

minimize:

The analysis is:

equivalent expression:

Case N=1

Case

Perfect observations:

x=Ea

E=[ e1 e1 e N ]

xa=x f

x

2J=aT−1 a[H x f Ea−d ]T R−1[H x f Ea−d ]

xa=x f

eHe T R−1d

−1He T R−1

He

R=o2 I xa

=x fe

He T d

2He T He

2=o

2

xa=x f

eHe T d

He T He

xa=x f

E[ −1H E

T R−1H E ]

−1HE

T R−1d

xa=x f

E H ET[ H E H E

TR ]

−1d

Sequential AUS

Unstable structures are identified as local extrema in the bred vector fields.

Horizontal localization is obtained by means of a modulating Gaussian function

The observed component He of the must be large and locally correlated with the innovation d= yo - H(xf)

Each structure is used to compute the analysis update in a sequential way: after using each structure, the updated state is used as a background field for the next analysis (next structure).

BDAS (Breeding on the Data Assimilation System) requires analysis-updating the perturbed states as well as the control state.

Exit when acceptable structures are not found anymore

Radar reflectivity (dBZ) estimated for the true state at level 15 (~700 hPa) at 03h00 as a function

of pressure, virtual temperature, and concentrations of precipitating water phases: rain, snow, hail.

data assimilation experiment Observations: radar reflectivity; analysis at 03h00, 04h00, 05h00, 06h00

RMS (root-mean-square) errors in time (h) for TEMPERATURE (°C) at levels: 05 (~925hPa), 08 (~850 hPa),

15 (~700 hPa), 25 (~500 hPa), 35 (~300hPa). Each color corresponds to a level. For each level there is just

one curev until 03h00, the first analysis time. After 03h00, filled squares reefer to errors of the freely evolving

trajectory; asterisks refer to the trajectory of the assimilation experiment.

RMS (root-mean-square) errors in time (h) for PRESSURE (Pa) at levels: 05 (~925hPa), 08 (~850 hPa), 15

(~700 hPa), 25 (~500 hPa), 35 (~300hPa). Each color corresponds to a level. For each level there is just one

curev until 03h00, the first analysis time. After 03h00, filled squares reefer to errors of the freely evolving

trajectory; asterisks refer to the trajectory of the assimilation experiment.

RMS (root-mean-square) errors in time (h) for SPECIFIC HUMIDITY (kg kg-1) at levels: 05 (~925hPa), 08

(~850 hPa), 15 (~700 hPa), 25 (~500 hPa), 35 (~300hPa). Each color corresponds to a level. For each level

there is just one curev until 03h00, the first analysis time. After 03h00, filled squares reefer to errors of the

freely evolving trajectory; asterisks refer to the trajectory of the assimilation experiment.

RMS (root-mean-square) errors in time (h) for WIND (horizontal components of velocity vector, m s-1) at levels:

05 (~925hPa), 08 (~850 hPa), 15 (~700 hPa), 25 (~500 hPa), 35 (~300hPa). Each color corresponds to a level.

For each level there is just one curev until 03h00, the first analysis time. After 03h00, filled squares reefer to

errors of the freely evolving trajectory; asterisks refer to the trajectory of the assimilation experiment.

RMS (root-mean-square) errors in time (h) for VERTICAL VELOCITY (vertical component of velocity vector, m

s-1) at levels: 05 (~925hPa), 08 (~850 hPa), 15 (~700 hPa), 25 (~500 hPa), 35 (~300hPa). Each color

corresponds to a level. For each level there is just one curev until 03h00, the first analysis time. After 03h00,

filled squares reefer to errors of the freely evolving trajectory; asterisks refer to the trajectory of the

assimilation experiment.

AUS references:

Carrassi, A., L. Descamps, O. Talagrand, A. Trevisan, F. Uboldi, 2008a. Controlling instabilities along a 3DVar analysis cycle by assimilating in the unstable subspace: a comparison with the EnKF. Nonlinear Processes in Geophysics, 15, 503-521.

Carrassi, A., M. Ghil, A. Trevisan, F. Uboldi, 2008b. Data assimilation as a nonlinear dynamical systems problem: stability and convergence of the prediction-assimilation system. Chaos, 18, 023112.

Carrassi, A., A. Trevisan, F. Uboldi, 2007. Adaptive observations and assimilation in the unstable subspace by breeding on the data-assimilation system. Tellus, 59A, 101-113.

Trevisan A., L. Palatella, 2010. The Extended Kalman Filter and its reduction to the unstable subspace. In Review.

Trevisan, A., M. D'Isidoro, O. Talagrand, 2010. Four-dimensional variational assimilation in the unstable subspace and the optimal subspace dimension. Quarterly Journal of the Royal Meteorological Society, 136, 487-496.

Trevisan, A., F. Uboldi, 2004. Assimilation of standard and targeted observations in the unstable subspace of the Observation-Analysis-Forecast cycle system. Journal of the Atmospheric Sciences, 61, 103-113.

Uboldi, F., A. Trevisan, 2006. Detecting unstable structures and controlling error growth by assimilation of standard and adaptive observations in a primitive equation ocean model. Nonlinear Processes in Geophysics, 13, 67-81.

Uboldi, F., A. Trevisan, A. Carrassi, 2005. Developing a dynamically based assimilation method for targeted and standard observations. Nonlinear Processes in Geophysics, 12, 149-156.

Contact: uboldi@magritte.it