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Dynamics and Structure of Three- Dimensional Error Covariance of a Mature Hurricane Jonathan Poterjoy and Fuqing Zhang Pennsylvania State University December 16, 2009
Dynamics and Structure of Three-Dimensional Error Covariance of a
Mature Hurricane
Jonathan Poterjoy and Fuqing Zhang Pennsylvania State University
December 16, 2009
The Tropical Cyclone Prediction Issue
A 1-2% per year increase in track forecasts has occurred over the last 3 decades (Franklin et al. 2003).
In general, little progress has been made in terms of intensity prediction.
WHY?
One deficiency comes from the use of isotropic, flow-independent background statistics in operational data assimilation systems, which are ill-suited for the highly flow-dependent background error covariances associated with tropical cyclones. Estimating non-static background statistics is a computationally expensive process, but may be required for the next generation of tropical cyclone prediction systems.
Experiment Description
This study examines the flow-dependent correlation structure of an axisymmetric vortex through a progression of simple to complex models:
• Rankine vortex model • Axisymmetric Rotunno-Emanuel (1987) maximum potential intensity model • Weather Research and Forecasting (WRF) model
For the Rankine vortex and axisymmetric experiments, random perturbations were added to initial conditions before numerical integration to create ensembles large enough for a reasonable estimation of forecast error. Correlations were calculated from a WRF ensemble of Hurricane Katrina forecasts for comparison.
Model Descriptions
Rankine Vortex
• Axisymmetric • Solid-body rotation for • 2-dimensional x-y plane • Variables: Vt, Vu, Vv, and Wz • Ensemble size: 500
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V t =
V 0
Rr, r ≤ R
V 0Rr, r ≥ R
⎧
⎨ ⎪
⎩ ⎪
€
r ≤ R
Model Descriptions
Rotunno-Emanuel (1987) Axisymmetric Vortex Model (RE87)
• Steady-state, axisymmetric hurricane simulation • Originally designed to study Maximum Potential Intensity (MPI) theory • Nonhydrostatic • Convection is accounted for explicitly • Initialized with a finite-amplitude vortex • Uses soundings from WRF Katrina members • Variables: Vt, Vu, Vv, Vw, dP, dT, Wz, Ql, and Qv • Domain:
• 45 x 45 x 15 • 9 km horizontal resolution • 1.2 km vertical resolution
• 2-dimensional x-z plane interpolated to 3-dimensions • 10 day lead-time • Ensemble size: 50
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ˆ z
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ˆ x
Model Descriptions Advanced Research WRF (ARW)
• Fully compressible, nonhydrostatic, mesoscale model • Variables: Vu, Vv, Vt, Vr, Vw, dP, dT, Wz, dT, QL, and QV • Nested vortex following domain:
• 253 x 253 x 34 • 4.5 km horizontal resolution
• Ensemble initialization: EnKF after assimilating hours of airborne Vr • 60 hour lead time ensemble of Hurricane Katrina forecasts • The forecast time of 1200 UTC August 28, 2005 was used:
• Katrina was at category 5 intensity at this time, making it easy to replicate using the RE87 model.
• Ensemble size: 92 (members which tracked over land were removed) • Storms re-centered based on location of minimum surface pressure • Domain reduced to 89 x 89 x 30 grid points • Variables averaged azimuthally
Tangential Velocity Profiles
Tang
entia
l Vel
ocity
(m/s
)
Radius (km)
RE87 Ensemble Members Mean and Standard Deviation
Cross Spatial Correlation: -Correlation between two
different forecast variables at different grid points.
Definitions
Cross Correlation: -Correlation between two
different forecast variables at the same grid point.
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Corr(x ijk,y lmn) =Exp[(x ijk−x )(y lmn−y )]
σ xσ y
=Cov(x ijk,y lmn)
σ xσ y
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x ≠ y, (i, j,k) = (l,m,n)
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x ≠ y, (i, j,k) ≠ (l,m,n)
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r(x11,y11) r(x12,y12) r(x13,y13)r(x 21,y 21) r(x 22,y 22) r(x 23,y 23)r(x 31,y 31) r(x 32,y 32) r(x 33,y 33)
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r(x 22,y11) r(x 22,y12) r(x 22,y13)r(x 22,y 21) r(x 22,y 22) r(x 22,y 23)r(x 22,y 31) r(x 22,y 32) r(x 22,y 33)
Tangential Velocity Comparison
Rankine Vortex ARW Katrina
Vt Standard Deviation
Vt Mean
Vt Standard Deviation
Vt Mean
Tangential Velocity Comparison RE87 t = 0
ARW ARW Avg
RE87 t = 10 d
Mean
Standard Deviation
Mean
Standard Deviation
Tangential Velocity Spatial Correlations
Rankine Vortex RE87 t = 0 RE87 t = 10 d
ARW Avg ARW
-Surface tangential velocity correlations with respect to a point marked by ‘c’.
RE87 t = 0 RE87 t = 10 d
ARW Avg ARW
-Surface Vu-Vv cross spatial correlations with respect to a point marked by ‘c’.
Vu-Vv Velocity Spatial Correlations
Rankine Vortex
Three-Dimensional Correlation Structures
Variables: U and V components of wind
Location: Surface point, ~90 km from vortex center
RE87 t = 10 d
ARW Avg ARW
Cross Correlations
Cross Spatial Correlations
Cross Correlations
Cross Spatial Correlations
Three-Dimensional Correlation Structures
Variables: Perturbation Temperature and radial velocity
Location: ~12 km vertical, ~80 km from vortex center
RE87 t = 10 d
ARW Avg ARW
Cross Correlations
Cross Spatial Correlations
Cross Correlations
Cross Spatial Correlations
Three-Dimensional Correlation Structures
Variables: Perturbation Temperature and vertical velocity
Location: ~9 km vertical ~90 km from vortex center
RE87 t = 10 d
ARW Avg ARW
Cross Correlations
Cross Spatial Correlations
Cross Correlations
Cross Spatial Correlations
Three-Dimensional Correlation Structures RE87 t = 10 d
ARW Avg ARW
Cross Correlations
Cross Spatial Correlations
Cross Correlations
Cross Spatial Correlations
Variables: Water vapor mixing ratio and vertical velocity
Location: ~11 km vertical ~18 km from vortex center
Three-Dimensional Correlation Structures RE87 t = 10 d
ARW Avg ARW
Cross Correlations
Cross Spatial Correlations
Cross Correlations
Cross Spatial Correlations
Variables: Perturbation pressure and perturbation temperature
Location: ~6 km vertical ~9 km from vortex center
Summary and Conclusion We examined the flow-dependent structure of forecast error from an
axisymmetric vortex through a progression of simple to complex models:
Rankine vortex Rotunno-Emanual MPI model ARW Katrina
• Synoptic scale correlation structures estimated from all three models were highly anisotropic and consistent with the underlying model dynamics.
• When the two lower order models were tuned to fit Katrina forecasts, i.e. in terms of maximum tangential wind speed and radius of maximum winds, they provided dynamically similar correlation structures.
• In fact, even with no changes made to the model dynamics, the RE87 model was able to resolve many of the same three-dimensional relationships observed with the WRF ensemble.
• Results from our experiments raise the question of whether or not a low-order axisymmetric vortex model can be used to estimate flow-dependant background error covariance for future tropical cyclone prediction systems.
