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A Class of Conservative Fourth-Order Advection Schemes and Impact of Enhanced Formal Accuracy on Extended-Range Forecasts

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  • 1 National Centers for Environmental Prediction, Camp Springs, Maryland
  • | 2 Alfred Wegener Institute, Bremerhaven, Germany
  • | 3 University Corporation for Atmospheric Research, Camp Springs, Maryland
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Abstract

Starting from three Eulerian second-order nonlinear advection schemes for semi-staggered Arakawa grids B/E, advection schemes of fourth order of formal accuracy were developed. All three second-order advection schemes control the nonlinear energy cascade in case of nondivergent flow by conserving quadratic quantities. Linearization of all three schemes leads to the same second-order linear advection scheme. The second-order term of the truncation error of the linear advection scheme has a special form so that it can be eliminated by modifying the advected quantity while still preserving consistency. Tests with linear advection of a cone confirm the advantage of the fourth-order scheme. However, if a localized, large amplitude and high wavenumber pattern is present in initial conditions, the clear advantage of the fourth-order scheme disappears.

The new nonlinear fourth-order schemes are quadratic conservative and reduce to the Arakawa Jacobian for advected quantities in case of nondivergent flow. In case of general flow the conservation properties of the new momentum advection schemes impose stricter constraint on the nonlinear cascade than the original second-order schemes. However, for nondivergent flow, the conservation properties of the fourth-order schemes cannot be proven in the same way as those of the original second-order schemes. Therefore, demanding long-term and low-resolution nonlinear tests were carried out in order to investigate how well the fourth-order schemes control the nonlinear energy cascade. All schemes were able to maintain meaningful solutions throughout the test.

Finally, the impact was examined of the fourth-order momentum advection on global medium-range forecasts. The 500-hPa anomaly correlation coefficient obtained using the best performing fourth-order scheme did not show an improvement compared to the tests using its second-order counterpart.

Corresponding author address: Zavisa Janjic, NCEP, 5200 Auth Rd., Camp Springs, MD 20746. E-mail: zavisa.janjic@noaa.gov

Abstract

Starting from three Eulerian second-order nonlinear advection schemes for semi-staggered Arakawa grids B/E, advection schemes of fourth order of formal accuracy were developed. All three second-order advection schemes control the nonlinear energy cascade in case of nondivergent flow by conserving quadratic quantities. Linearization of all three schemes leads to the same second-order linear advection scheme. The second-order term of the truncation error of the linear advection scheme has a special form so that it can be eliminated by modifying the advected quantity while still preserving consistency. Tests with linear advection of a cone confirm the advantage of the fourth-order scheme. However, if a localized, large amplitude and high wavenumber pattern is present in initial conditions, the clear advantage of the fourth-order scheme disappears.

The new nonlinear fourth-order schemes are quadratic conservative and reduce to the Arakawa Jacobian for advected quantities in case of nondivergent flow. In case of general flow the conservation properties of the new momentum advection schemes impose stricter constraint on the nonlinear cascade than the original second-order schemes. However, for nondivergent flow, the conservation properties of the fourth-order schemes cannot be proven in the same way as those of the original second-order schemes. Therefore, demanding long-term and low-resolution nonlinear tests were carried out in order to investigate how well the fourth-order schemes control the nonlinear energy cascade. All schemes were able to maintain meaningful solutions throughout the test.

Finally, the impact was examined of the fourth-order momentum advection on global medium-range forecasts. The 500-hPa anomaly correlation coefficient obtained using the best performing fourth-order scheme did not show an improvement compared to the tests using its second-order counterpart.

Corresponding author address: Zavisa Janjic, NCEP, 5200 Auth Rd., Camp Springs, MD 20746. E-mail: zavisa.janjic@noaa.gov
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