Putting Moisture into Spectral Models

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Putting Moisture into Spectral Models

• November 7, 2002
• AT 703
• Jonathan Vigh

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A transport scheme should have the following
properties

• Accuracy
• Stability
• Transportive
• Local
• Conservative
• Shape-Preserving
• Computationally affordable

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Overview of advection in a spectral model.
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Advantages of Spectral Methods

• Well-suited to spherical geometry (no pole
  problem, especially with ? truncation).
• No approximation in horizontal derivatives (phase
  speeds of linear waves, advection are exact).
• Convergence of smooth functions is fast with only
  a few modes.
• Allows easy use of semi-implicit time
  differencing, leading to cost savings.

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Disadvantages of Spectral Methods

• Nothing is exactly conserved, not even mass.
• Partly because of failure to conserve the
  mass-weighted total energy, artificial damping is
  needed to maintain computational stability.
• Because of spectral truncation in the transform
  method, physical parameterizations do not always
  have the intended effect.
• At high resolution, spectral methods are
  computationally expensive compared to grid point
  models.
• Spectral models suffer from Gibbs Phenomenon.
• Moisture advection using spectral methods gives
  poor results.

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Gibbs Phenomenon

• Gibbs Phenomenon is a spurious oscillation that
  occurs when using a truncated eigenfunction
  series (such as Fourier, Chebyshev, or Legendre
  series) at a simple discontinuity.
• It is characterized by an initial overshoot, then
  a pattern of undershoot, overshoot oscillations
  that decrease in amplitude further from the
  physical discontinuity.

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Gibbs Phenomenon in Spectral Models

• Truncation of an eigenfunction series expansion
  of a discontinuous function produces spurious
  waves in the vicinity of sharp gradients.
• Effect may occur for any spectral variable which
  must be represented in physical space (i.e.
  topography and moisture).
• Especially bad for topography, water vapor, and
  condensate (cloudiness).

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Effects of Gibbs Oscillations

• Variance is introduced in model variables that do
  not correspond to any real physical cause.
• Can influence basic time-mean fields such as
  surface pressure and meridional wind, affecting
  simulation of phenomenon such as El Niño.
• Can affect surface fluxes.
• Can interact with and cause problems with
  physical parameterizations.
• Cloudiness and precipitation are adversely
  affected.

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Spectral moisture transport

• Moisture varies by several orders of magnitude
  between the surface and the stratosphere, and
  between the EQ and the poles.
• Thus, moisture advection suffers severe
  representation error in spectral models.
• Negative water is produced in undershoot areas,
  and excess water appears in overshoot areas.
• Production of negative water can be as much as
  20 of the globally averaged precipitation rate.

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Moisture headaches

• Negative values of any tracer are bad, but the
  problem is particularly troublesome with moisture
  due to thermodynamic interaction with the
  dynamics through clouds and precipitation.
• Latent heat release associated with condensation
  is related to the absolute changes of water
  vapor.
• Clouds and radiation are influenced more by the
  relative variations in water vapor.
• In order to correctly simulate climate, both of
  these processes must be accurate.

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Ways to mitigate moisture problems

• Use a different moisture variable.
• Fill the holes.
• Use a transformed variable which cant go
  negative.
• Dont use spectral methods for moisture transport
  (do it on the grid).

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Are there any moisture variables that arent as
susceptible?

• Choices of moisture variables to predict
• Dewpoint depression
• Mixing ratio
• Relative humidity
• Other choices are similar specific humidity,
  vapor pressure, and dewpoint temperature

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Which Moisture Variable?

• Ian Simmonds (1975) did some experiments to
  assess the relative accuracy of the spectral
  representation of these moisture variables in
  being able to reproduce grid-point values of
  these quantities.
• Defined information content for the finite
  spherical harmonic series approximation, tested
  for various N-wave truncations.
• Found that mixing ratio was a poor choice, dew
  point depression was best, slightly better than
  relative humidity.

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Why not just fill the holes?

• One solution (commonly used) is to simply fill
  the holes of negative water.
• Borrow either from a neighboring grid cell or
  from the global mean.
• Local borrowing is arbitrary and can be
  expensive.
• Global borrowing is easy, but not physical.
• Can bother the parameterizations.
• Other solutions locally adaptive FCT scheme

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Use of a transformed moisture variable

• Ooyama (2001) suggests use of a predictand that
  cannot become negative a transformed moisture
  variable
• As an example, instead of predicting µ, you
  could predict the transformed variable
• for which the recovered µexp v will always be
  positive.
• The logarithmic function bends µ at all values.
  The mean of µ is not well preserved.

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Try a hyperbolic transform
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Try a biased hyperbolic transform
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• This allows the predictand ? to go somewhat
  negative (without an adverse effect on µ), while
  preserving the proper shape of µ out at values
  not close to zero.
• Effect of µ0 on the potential temperature
  difference is less than 3 x 10-4 K.
• No discernible difference in simulations for µ0
  10-7 or 10-8. A slight difference for
  µ0 10-6.

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If moisture advection doesnt work in the
spectral method, dont use it.

• Most modern spectral models handle the transport
  of moisture (and other tracers) in gridpoint
  space using finite-difference methods.
• The dry dynamics (linear advection, linear waves,
  PGF terms) are handled spectrally.
• Moisture and other tracers are advected on the
  grid.
• All physical parameterizations are also done on
  the grid.

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Lagrangian Techniques

• In the absence of sources and sinks, mixing ratio
  is constant along a trajectory.
• So the value of the tracer at the end of the
  trajectory should be the same as at the beginning
  of trajectory.
• All that is needed is to calculate the
  trajectory, then determine the value of the
  tracer at the beginning of the trajectory.

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Semi-Lagrangian Transport (SLT)

• Trajectories are calculated from one time step
  only with no imposed continuity from time step to
  time step.
• Forecast is desired at a set of gridpoints
  referred to as arrival points at time t ?T.
• Trajectories are calculated backwards from the
  arrival points at time t ?T to time t, where
  they define a set of departure points.
• Values at departure points are determined by
  interpolation from the background grid.

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Caveats

• A straightforward SLT scheme requires some sort
  of interpolation to get departure values.
• A simple interpolation scheme makes SLT similar
  to any other forward time scheme which suffers
  from excessive diffusion.
• SLT suffers from strong flow over the pole.
• In practice, another grid (such as geodesic) is
  used in the vicinity of the pole.
• Conservation is not attained mass fixer is
  still required (but it can be tied to the just
  the advection processes, unlike in the spectral
  method)

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Shape-preserving SLT Schemes

• Trajectory is divided into two parts
• One goes from the arrival point to the gridpoint
  closest to the departure point
• The other goes from that gridpoint to the
  departure point.
• First segment between gridpoints is treated in
  Lagrangian fashion, with no interpolation
  required.
• Second segment is treated in an Eulerian fashion,
  but now the CFL condition is always satisfied
  (the segment stays within one grid interval).

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Conclusions

• Many approaches for transport on the sphere have
  been tried.
• There is no ideal transport scheme in spherical
  geometry, although some are clearly more
  attractive than others.
• Shape-preserving SLT schemes seem to be the best
  choice at the moment.
• In modeling, there are always trade-offs. Silver
  bullets are rare, and it requires careful
  analysis to design a model with the properties
  that you desire.

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References

• Arfkin, George, 1985 Mathematical Methods for
  Physicists Third Edition. Academic Press, 985
  pp.
• Gottlieb, D. and S. A. Orszag, 1977 Numerical
  Analysis of Spectral Methods Theory and
  Applications. Society for Industrial and Applied
  Mathematics, CBMS-NSF Series, No. 26, 170 pp.
• Lindberg, C. and A. J. Broccoli, 1996
  Representation of Topography in Spectral Climate
  Models and its Effect on Simulated Precipitation.
  J. of Climate, 9, 2641-2659.
• Ooyama, K. V., 2001 A Dynamic and Thermodynamic
  Foundation for Modeling the Moist Atmosphere with
  Parameterized Microphysics. J. Atmos. Sci., 58,
  2073-2102.
• Rasch, P. J. and D. L. Williamson, 1990
  Computational Aspects of Moisture Transport in
  Global Models of the Atmosphere. Q. J. R.
  Meteorol. Soc., 116,1071-1090.
• Randall, D., 2001 Spherical Methods. AT 604
  Notes, 249-270.
• Schepetkin, A. F. and J. C. McWilliams, 1998
  Quasi-Monotone Advection Schemes Based on
  Explicit Locally Adaptive Dissipation. Mon. Wea.
  Rev., 126, 1541-1580.
• Simmonds, Ian, 1975 The Spectral Representation
  of Moisture. J. Appl. Meteor., 14,175-179.
• Williamson, D. L., 1992 Review of Numerical
  Approaches for Modeling Global Transport. Air
  Pollution Modeling and its Applications IX, Ed.
  H. van Dop and G. Kallos, Plenum Press, New York.
  
• ---, and P. J. Rasch, 1994 Water Vapor Transport
  in the NCAR CCM2. Tellus, 46A, 34-51.