• Bleck, R., 1974: Short range prediction in isentropic coordinates with filtered and unfiltered numerical models. Mon. Wea. Rev., 102 , 813829.

    • Search Google Scholar
    • Export Citation
  • Dritschel, D. G., and M. H. P. Ambaum, 1997: A contour-advective semi-Lagrangian algorithm for the simulation of fine-scale conservative fields. Quart. J. Roy. Meteor. Soc., 123 , 10971130.

    • Search Google Scholar
    • Export Citation
  • Dritschel, D. G., and A. R. Mohebalhojeh, 2000: The contour-advective semi-Lagrangian algorithm: Keeping the balance. Proc. Developments in Numerical Methods for Very High Resolution Global Models, Reading, United Kingdom, European Centre for Medium-Range Weather Forecasts, 119–136. [Available online at http://www.ecmwf.int/publications/library/ecpublications/_pdf/workshop/2000/HighResolution/ws_highres_dritschel.pdf.].

  • Dritschel, D. G., L. M. Polvani, and A. R. Mohebalhojeh, 1999: The contour-advective semi-Lagrangian algorithm for the shallow water equations. Mon. Wea. Rev., 127 , 15511565.

    • Search Google Scholar
    • Export Citation
  • Hasselmann, K., 1976: Stochastic climate models. Part I. Theory. Tellus, 28 , 473485.

  • McIntyre, M. E., and W. A. Norton, 2000: Potential vorticity inversion on a hemisphere. J. Atmos. Sci., 57 , 12141235. Corrigendum. 58 , 949.

    • Search Google Scholar
    • Export Citation
  • Mohebalhojeh, A. R., 2002: On shallow-water potential-vorticity inversion by Rossby number expansions. Quart. J. Roy. Meteor. Soc., 128 , 679694.

    • Search Google Scholar
    • Export Citation
  • Mohebalhojeh, A. R., and D. G. Dritschel, 2000: On the representation of gravity waves in numerical models of the shallow water equations. Quart. J. Roy. Meteor. Soc., 126 , 669688.

    • Search Google Scholar
    • Export Citation
  • Mohebalhojeh, A. R., and D. G. Dritschel, 2001: Hierarchies of balance conditions for the f-plane shallow water equations. J. Atmos. Sci., 58 , 24112426.

    • Search Google Scholar
    • Export Citation
  • Mohebalhojeh, A. R., and D. G. Dritschel, 2004: Contour-advective semi-Lagrangian algorithms for many-layer primitive equation models. Quart. J. Roy. Meteor. Soc., 130 , 347364.

    • Search Google Scholar
    • Export Citation
  • Mohebalhojeh, A. R., and M. E. McIntyre, 2007: Local mass conservation and velocity splitting in PV-based balanced models. Part I: The hyperbalance equations. J. Atmos. Sci., 64 , 17821793.

    • Search Google Scholar
    • Export Citation
  • Rossby, C. G., 1936: Dynamics of steady ocean currents in the light of experimental fluid mechanics. Pap. Phys. Oceanogr. Meteor., 5 , 143.

    • Search Google Scholar
    • Export Citation
  • Temperton, C., and A. Staniforth, 1987: An efficient two-time-level semi-Lagrangian semi-implicit integrating scheme. Quart. J. Roy. Meteor. Soc., 113 , 10251040.

    • Search Google Scholar
    • Export Citation
  • Warn, T., O. Bokhove, T. G. Shepherd, and G. K. Vallis, 1995: Rossby number expansions, slaving principles, and balance dynamics. Quart. J. Roy. Meteor. Soc., 121 , 723739.

    • Search Google Scholar
    • Export Citation
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 36 9 0
PDF Downloads 14 7 0

Local Mass Conservation and Velocity Splitting in PV-Based Balanced Models. Part II: Numerical Results

Ali R. MohebalhojehSchool of Mathematics and Statistics, University of St Andrews, St Andrews, United Kingdom, and Institute of Geophysics, University of Tehran, Tehran, Iran

Search for other papers by Ali R. Mohebalhojeh in
Current site
Google Scholar
PubMed
Close
and
Michael E. McIntyreCentre for Atmospheric Science,* Department of Applied Mathematics and Theoretical Physics, Cambridge University, Cambridge, United Kingdom

Search for other papers by Michael E. McIntyre in
Current site
Google Scholar
PubMed
Close
Restricted access

Abstract

The effects of enforcing local mass conservation on the accuracy of non-Hamiltonian potential-vorticity- based balanced models (PBMs) are examined numerically for a set of chaotic shallow-water f-plane vortical flows in a doubly periodic square domain. The flows are spawned by an unstable jet and all have domain-maximum Froude and Rossby numbers Fr ∼0.5 and Ro ∼1, far from the usual asymptotic limits Ro → 0, Fr → 0, with Fr defined in the standard way as flow speed over gravity wave speed. The PBMs considered are the plain and hyperbalance PBMs defined in Part I. More precisely, they are the plain-δδ, plain-γγ, and plain-δγ PBMs and the corresponding hyperbalance PBMs, of various orders, where “order” is related to the number of time derivatives of the divergence equation used in defining balance and potential-vorticity inversion. For brevity the corresponding hyperbalance PBMs are called the hyper-δδ, hyper-γγ, and hyper-δγ PBMs, respectively. As proved in Part I, except for the leading-order plain-γγ each plain PBM violates local mass conservation. Each hyperbalance PBM results from enforcing local mass conservation on the corresponding plain PBM. The process of thus deriving a hyperbalance PBM from a plain PBM is referred to for brevity as plain-to-hyper conversion. The question is whether such conversion degrades the accuracy, as conjectured by McIntyre and Norton.

Cumulative accuracy is tested by running each PBM alongside a suitably initialized primitive equation (PE) model for up to 30 days, corresponding to many vortex rotations. The accuracy is sensitively measured by the smallness of the ratio ϵ = ||QPBMQPE||2/||QPE||2, where QPBM and QPE denote the potential vorticity fields of the PBM and the PEs, respectively, and || ||2 is the L2 norm. At 30 days the most accurate PBMs have ϵ ≈ 10−2 with PV fields hardly distinguishable visually from those of the PEs, even down to tiny details. Most accurate is defined by minimizing ϵ over all orders and truncation types δδ, γγ, and δγ. Contrary to McIntyre and Norton’s conjecture, the minimal ϵ values did not differ systematically or significantly between plain and hyperbalance PBMs. The smallness of ϵ suggests that the slow manifolds defined by the balance relations of the most accurate PBMs, both plain and hyperbalance, are astonishingly close to being invariant manifolds of the PEs, at least throughout those parts of phase space for which Ro ≲ 1 and Fr ≲ 0.5.

As another way of quantifying the departures from such invariance, that is, of quantifying the fuzziness of the PEs’ slow quasimanifold, initialization experiments starting at days 1, 2, . . . 10 were carried out in which attention was focused on the amplitudes of inertia–gravity waves representing the imbalance arising in 1-day PE runs. With balance defined by the most accurate PBMs, and imbalance by departures therefrom, the results of the initialization experiments suggest a negative correlation between early imbalance and late cumulative error ϵ. In such near-optimal conditions the imbalance seems to be acting like weak background noise producing an effect analogous to so-called stochastic resonance, in that a slight increase in noise level brings PE behavior closer to the balanced behavior defined by the most accurate PBMs when measured cumulatively over 30 days.

* The Centre for Atmospheric Science is a joint initiative of the Department of Chemistry and the Department of Applied Mathematics and Theoretical Physics (more information available online at www.atm.damtp.cam.ac.uk/)

Corresponding author address: A. R. Mohebalhojeh, School of Mathematics and Statistics, University of St Andrews, North Haugh, St Andrews KY16 9SS, United Kingdom. Email: arm@mcs.st-and.ac.uk

Abstract

The effects of enforcing local mass conservation on the accuracy of non-Hamiltonian potential-vorticity- based balanced models (PBMs) are examined numerically for a set of chaotic shallow-water f-plane vortical flows in a doubly periodic square domain. The flows are spawned by an unstable jet and all have domain-maximum Froude and Rossby numbers Fr ∼0.5 and Ro ∼1, far from the usual asymptotic limits Ro → 0, Fr → 0, with Fr defined in the standard way as flow speed over gravity wave speed. The PBMs considered are the plain and hyperbalance PBMs defined in Part I. More precisely, they are the plain-δδ, plain-γγ, and plain-δγ PBMs and the corresponding hyperbalance PBMs, of various orders, where “order” is related to the number of time derivatives of the divergence equation used in defining balance and potential-vorticity inversion. For brevity the corresponding hyperbalance PBMs are called the hyper-δδ, hyper-γγ, and hyper-δγ PBMs, respectively. As proved in Part I, except for the leading-order plain-γγ each plain PBM violates local mass conservation. Each hyperbalance PBM results from enforcing local mass conservation on the corresponding plain PBM. The process of thus deriving a hyperbalance PBM from a plain PBM is referred to for brevity as plain-to-hyper conversion. The question is whether such conversion degrades the accuracy, as conjectured by McIntyre and Norton.

Cumulative accuracy is tested by running each PBM alongside a suitably initialized primitive equation (PE) model for up to 30 days, corresponding to many vortex rotations. The accuracy is sensitively measured by the smallness of the ratio ϵ = ||QPBMQPE||2/||QPE||2, where QPBM and QPE denote the potential vorticity fields of the PBM and the PEs, respectively, and || ||2 is the L2 norm. At 30 days the most accurate PBMs have ϵ ≈ 10−2 with PV fields hardly distinguishable visually from those of the PEs, even down to tiny details. Most accurate is defined by minimizing ϵ over all orders and truncation types δδ, γγ, and δγ. Contrary to McIntyre and Norton’s conjecture, the minimal ϵ values did not differ systematically or significantly between plain and hyperbalance PBMs. The smallness of ϵ suggests that the slow manifolds defined by the balance relations of the most accurate PBMs, both plain and hyperbalance, are astonishingly close to being invariant manifolds of the PEs, at least throughout those parts of phase space for which Ro ≲ 1 and Fr ≲ 0.5.

As another way of quantifying the departures from such invariance, that is, of quantifying the fuzziness of the PEs’ slow quasimanifold, initialization experiments starting at days 1, 2, . . . 10 were carried out in which attention was focused on the amplitudes of inertia–gravity waves representing the imbalance arising in 1-day PE runs. With balance defined by the most accurate PBMs, and imbalance by departures therefrom, the results of the initialization experiments suggest a negative correlation between early imbalance and late cumulative error ϵ. In such near-optimal conditions the imbalance seems to be acting like weak background noise producing an effect analogous to so-called stochastic resonance, in that a slight increase in noise level brings PE behavior closer to the balanced behavior defined by the most accurate PBMs when measured cumulatively over 30 days.

* The Centre for Atmospheric Science is a joint initiative of the Department of Chemistry and the Department of Applied Mathematics and Theoretical Physics (more information available online at www.atm.damtp.cam.ac.uk/)

Corresponding author address: A. R. Mohebalhojeh, School of Mathematics and Statistics, University of St Andrews, North Haugh, St Andrews KY16 9SS, United Kingdom. Email: arm@mcs.st-and.ac.uk

Save