An idealised fluid model for inexpensive DA experiments ... · uid model for inexpensive DA...

An idealised fluid model for inexpensive DAexperiments and its relevance for NWP

Tom Kent*, Onno Bokhove, Steve Tobias

Dept. of Applied MathsUniversity of Leeds, UK

*Email: [email protected]

10th Adjoint Workshop, West Virginia1-5 June 2015

Tom Kent University of Leeds

Modified SW model for DA

DA: from large- to convective-scale

High-resolution (convective-scale) NWP models are becoming the norm

I more dynamical processes such as convection, cloud formation, andsmall-scale gravity waves, are resolved explicitly

DA techniques need to evolve in order to keep up with the developmentsin high-resolution NWP

I breakdown of dynamical balances (e.g., hydrostatic andsemi/quasi-gestrophic) at smaller scales

I strongly nonlinear processes associated with convection andmoisture/precipitation

I move towards ensemble-based methods



Using idealised models

It may be unfeasible, and indeed undesirable, to investigate the potentialof DA schemes on state-of-the-art NWP models. Instead idealised modelscan be employed that:

I capture some fundamental processes

I are computationally inexpensive to implement

I allow an extensive investigation of a forecast/assimilation system ina controlled environment

‘Toy’ models:

I Lorenz (L63, L95, L2005, ... )

I BV/QG models (Bokhove et al., poster this workshop)

I simplified NWP models



Using idealised models: approach

1. Describe a physically plausible idealised model and implementnumerically.

I based on the shallow water equations (SWEs).I compare dynamics of the modified model to those of the classical

shallow water theory

2. Ensemble-based DA - relevant for convective-scale NWP?I initial perturbations to represent forecast errorI “tuning” the observing system and the observational influence

diagnostic

3. Current/future work and ideas.I DA: a comparison with VARI advanced numerics: non-negativity of ‘rain’I other fluid dynamical modelsI which characteristics of NWP can we seek to replicate in idealised

models?



1. SWEs: an extension

Aim: modify the SWEs to include more complex dynamics relevant for the‘convective-scale’, extending the model employed by Wursch and Craig (2014).

I convective updrafts - artificially mimic conditional instability (positive buoyancy)

I idealised representation of precipitation, including source and sink.

I contain switches for the onset of convection and precipitation - realistic (andhighly nonlinear) features of operational NWP models.

2D rotating SWEs on an f -plane with no variation

in the y-direction (∂y = 0):

∂th+ ∂x(hu) = 0,

∂t(hu) + ∂x(hu2 + p(h))− fhv = −gh∂xb,

∂t(hv) + ∂x(huv) + fhu = 0,

∂tb = 0,

where p(h) is an effective pressure: p(h) = 12gh2.



Modified SWEs

Ingredients:

I two threshold heights Hc < Hr: when fluid exceeds these heights, differentmechanisms kick in and alter the classical SW dynamics.

I modifications to the effective pressure gradient (equivalently, geopotentialgradient) in the momentum equation.

I extra equation for the conservation of model ‘rain’ to close the system.

∂th+ ∂x(hu) = 0,

∂t(hu) + ∂x(hu2 + p(h)) + hc20∂xr − fhv = −gh∂xb,

∂t(hv) + ∂x(huv) + fhu = 0,

∂t(hr) + ∂x(hur) + hβ∂xu+ αhr = 0,

∂tb = 0,

where p(h) =

{12gH2

c , for h+ b > Hc,12gh2, otherwise,

and β =

{β, for h+ b > Hr, ∂xu < 0,

0, otherwise.



Some theoretical aspects

I Shallow water systems are hyperbolic, and can thus be solved via a range ofnumerical recipes for hyperbolic syststems. What about the modified system?

I Vector formulation:

∂tU + ∂xF(U) + G(U)∂xU + S(U) = 0

I Hyperbolicity determined by eigenstructure (all eigenvalues must be real).Eigenvalues of the system are determined by the matrix:

∂F/∂U + G(U) =

0 1 0 0 0

−u2 − c20r + ∂hp 2u c20 0 gh

−u(β + r) β + r u 0 0−uv v 0 u 00 0 0 0 0

.

I This matrix has five eigenvalues:

λ1,2 = u±√∂hp+ c20β, λ3,4 = u, and λ5 = 0,

I Since p(h) is non-decreasing and β non-negative, the eigenvalues are real.Hence, the modified SW model is hyperbolic.



Numerics

Scheme:

I large literature on numerical routines for hyperbolic systems of PDEs.

I Rhebergen et al. (2008) developed a novel discontinuous Galerkin (DG) finiteelement framework for hyperbolic system of PDEs with non-conservativeproducts G(U)∂xU.

I in most simple case (DG0), analagous to Godunov’s FV scheme in which anumerical flux must be evaluated

d

dtUk +

1

4xk

[PNC(Uk,Uk+1)− PNC(Uk−1,Uk)

]+

S(Uk)

4xk= 0.

Experiments:

I Rossby geostrophic adjustment in a periodic domain

I describes the evolution of the free surface height h when disturbed from its reststate by a transverse jet, i.e., fluid with an initial constant height profile issubject to a localised v-velocity distribution.

I non-dimensional parameters: Ro = 1 and Fr = 2.



Adjustment of a transverse jet in RSW

Below Hc and Hr:

−5 0 5

0.51.5

t=0

t=0.1T

t=0.2T

t=0.3T

t=0.4T

t=0.5T

x

h/H

x

t

h(x,t)/H

−5 0 5

0.1

0.2

0.3

0.4

0.5

1

2

3

4

5

x

t

r(x,t)

−5 0 5

0.1

0.2

0.3

0.4

0.5

0

0.1

0.2

0.3

0.4

0.5

Above Hc but below Hr:

−5 0 5

0.51.5

t=0

t=0.1T

t=0.2T

t=0.3T

t=0.4T

t=0.5T

x

h/H

x

t

h(x,t)/H

−5 0 5

0.1

0.2

0.3

0.4

0.5

1

2

3

4

5

x

t

r(x,t)

−5 0 5

0.1

0.2

0.3

0.4

0.5

0

0.1

0.2

0.3

0.4

0.5

Above Hc and Hr:

−5 0 5

0.51.5

t=0

t=0.1T

t=0.2T

t=0.3T

t=0.4T

t=0.5T

x

h/H

x

t

h(x,t)/H

−5 0 5

0.1

0.2

0.3

0.4

0.5

1

2

3

4

5

xt

r(x,t)

−5 0 5

0.1

0.2

0.3

0.4

0.5

0

0.1

0.2

0.3

0.4

0.5



2. Ensemble-based DA for idealised models

Ensemble Kalman filter: twin model setting

I Imperfect model:I “truth” trajectory: run at high resolutionI “forecast” model: run at lower resolution at which small-scale

features (e.g., localised moisture transport) are not fully resolvedI ensemble (covariance) inflation (xf

i ← γ(xfi − xf ) + xf ) applied to

account for the model error due to resolution mismatchI localisation (Pf ← ρloc ◦Pf ) applied to damp spurious long-range

correlations

I “tuning” the observing system: what to observe? how often? withhow much noise?

I observational influence diagnostic (after Cardinali et al. (2004))averaged over cycles:

OI =tr(HK)

p



Before assimilating...: ensemble spread as a representation of forecast error

0.00 0.02 0.04 0.06 0.08 0.100.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

h(x

,t)

RMS spread and RMS error of ensemble mean

Spread

Ens. mean RMSE

0.00 0.02 0.04 0.06 0.08 0.100.0

0.5

1.0

1.5

2.0

hu(x

,t)

0.00 0.02 0.04 0.06 0.08 0.100.0

0.5

1.0

1.5

2.0

hv(x,t)

0.00 0.02 0.04 0.06 0.08 0.10

t

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

hr(x,t)

Ratio of ensemble spread (N = 100) to forecasterror:

0.00 0.02 0.04 0.06 0.08 0.100

1

2

3

4

5

6

7

8

RMS spread / RMSE of mean h

hu

hv

hr

Mean



Cycled assimilation...: how does an analysis look?

0.0 0.2 0.4 0.6 0.8 1.0

0.60.81.01.21.41.6

h(x

)

Ensembles

Ens. mean

Obs.

Analysis

0.0 0.2 0.4 0.6 0.8 1.02.01.51.00.50.00.51.01.5

hu(x

)

0.0 0.2 0.4 0.6 0.8 1.0

x

3210123

hv(x)

0.0 0.2 0.4 0.6 0.8 1.0

x

0.0

0.2

0.4

0.6

0.8

1.0

hr(x)

Field-averaged RMS errors afteran analysis cycle (Obs. error =

0.1):

Forecast Analysish 0.0731 0.0725hu 0.1052 0.0812hv 0.1374 0.0696hr 0.0169 0.0238

Observational influence diagnostic:

OI =tr(HK)

p= 0.28



Cycled assimilation...: how does an analysis look?

0.0 0.2 0.4 0.6 0.8 1.0

0.60.81.01.21.41.6

h(x

)

Ensembles

Ens. mean

Obs.

Analysis

0.0 0.2 0.4 0.6 0.8 1.02.01.51.00.50.00.51.01.5

hu(x

)

0.0 0.2 0.4 0.6 0.8 1.0

x

3210123

hv(x)

0.0 0.2 0.4 0.6 0.8 1.0

x

0.0

0.2

0.4

0.6

0.8

1.0

hr(x)

Field-averaged RMS errors afteran analysis cycle (Obs. error =

0.05):

Forecast Analysish 0.0828 0.0816hu 0.0991 0.0906hv 0.1297 0.0793hr 0.0200 0.0293

Observational influence diagnostic:

OI =tr(HK)

p= 0.42



Lots of parameters and different set-ups to explore and play with:

I observe only one variable (e.g., the height field) and compare; orobserve nonlinearly (e.g., radial wind)

I include topography and observe downstream of a mountain

I increase the ratio of truth to forecast resolution to observesmaller-scale features

I (too) many more possibilities...



3. Current/future work and ideas

DA:

I setting up a demonstration system that compares EnKF with VAR in which Bmatrix is derived from ensemble.

Numerics:

I extension to ensure non-negativity of hr, a la Audusse et al., 2004.

PNC(Uk,Uk+1) −→ PNC(U(k+1/2)−,U(k+1/2)+)

I reconstructed states U(k+1/2)± impose that h and hr cannot become negative

yet dry states hr = 0 can be computed (given a derived time-step criterion).

Other models of interest:

I (dimensionally-reduced) adapted moist Boussinesq shallow water equations(after Zerroukat and Allen, 2015)

I 3D QG model with anisotropic rotating convection (Bokhove et al., poster)



3. Current/future work and ideas

Other diagnostics and the question of ‘relevance’:

I how can findings based on ‘toy’ models generalise to and provide useful insightfor operational NWP forecast/assimilation systems?

I observational influence diagnostic:I global NWP: 0.15 (Cardinali et al., 2004)I convective-scale NWP: 0.2 - 0.5? (Brousseau et al., 2014)

I error-growth properties of the idealised model should be similar to those in

operational models:I error-growth characteristics of assimilating model determine magnitude and

structure of the updated Pf represented by the ensemble.I error-doubling time for forecast error for global NWP known to be on the

order of days - what about convective-scale?



Summary and outlook

I novel fluid dynamical models to fill the ‘complexity gap’ betweenODE models and the primitive equations / state-of-the-art NWPmodels

I Idealised convective-scale DA experiments with characteristicsrelevant for NWP

I Implement a variational algorithm (in which initial covariance comesfrom the ensemble)

I Integrate model(s) into Met Office’s nascent ‘VarPy’ framework as arepository for idealised DA experiments



Thank you very much for your attention.

References:I Audusse et al., 2004: A fast and stable well-balanced scheme with hydrostatic

reconstruction for shallow water flows. SIAM JSC, 25(6), 2050-2065.I Bouchut, F., Sommer, J., and Zeitlin, V., 2004: Frontal geostrophic adjustment

and nonlinear wave phenomena in one-dimensional rotating shallow water. Part2. High-resolution numerical simulations. J. Fluid Mech., 514:35-63.

I Brousseau et al., 2014: A posteriori diagnostics of the impact of observations onthe AROME-France convective-scale data assimilation system. QJRMS,140(680), 982-994.

I Cardinali, C., Pezzulli, S., and Andersson, E, 2004: Influence-matrix diagnosticof a data assimilation system. QJRMS, 130(603):2767-2786.

I Rhebergen, S., Bokhove, O., and Van der Vegt, J., 2008: Discontinuous Galerkinfinite element methods for hyperbolic nonconservative partial differentialequations. J. Comp. Phys., 227(3), 1887-1922.

I Wursch, M., and Craig, G.C., 2014: A simple dynamical model of cumulusconvection for data assimilation research. Meteorologische Zeitschrift,23(5):483-490.

I Zerroukat, M., and Allen, T., 2015: A moist Boussinesq shallow water equationsset for testing atmospheric models. J. Comp. Phys., 290, 55-72.



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