Learning Center
Plans & pricing Sign in
Sign Out



									Forward-in-time Methods in
     Ocean Modeling

        Matthew Hecht
      Los Alamos National Lab
      Primitive Equation Ocean
• Incompressible Navier-Stokes Eqns, but with:
   – Hydrostatic pressure
   – Shallow depth, relative to rotationally-constrained
     horizontal scales
• Boussinesq approximation usually made
   – Ignore variations in density, , except in pressure
     gradient term
             Ocean model grids
• Horizontal grid is usually fixed, orthogonal
• Vertical grid can be:
   – Fixed vertical, independent of x, y and t (Z-grid)
   – Density, as in a stacked shallow water (isopycnal, or
     layered model)
   – Arbitrary Lagrangian/Eulerian (ALE)
 One more point: Modal decomposition

• Momentum equations separated into
  – vertically-averaged component (“barotropic”)
  – departure from the vertical average
    (“baroclinic” component)
• Fast surface gravity waves contained within
  equations for vertically-averaged flow
  – Time step for 3-D momentum equations can be
    something like 2 orders of magnitude larger
    than for the 2-D “barotropic” equations
   Who cares about the oceans?
  Climate scientists, oceanographers, coastal
        residents of South Wales, …
• Climate ocean models typically Order(1º)
   – Most of turbulent spectrum parameterized, not resolved
• Atmosphere forces the oceans
   – through winds, fluxes of heat, fresh water
• Ocean responds
   – with transport of heat, salinity
• Ocean forces the atmosphere
   – through sea surface temperature
• So… an ocean model for climate had better
  produce the right sea surface temperatures
     Stommel Gyre test problem
• Stommel (1948)
  found analytical
  solution for gyre
  circulation from
  balance of wind
  stress and bottom
  drag, when on a
  rotating sphere

 Passive tracer advective test problem in highly sheared
 flow field, from Hecht, Holland and Rasch
 (J. Geophys. Res. 1995)
           Reference Solution

We send the tracer
through the
western boundary
current once.

Reference solution
is produced
                Selected results
 As on previous slide        Centered leapfrog

With 1
antidiffusive                          O(3) upwind
       Centered cases with diffusion
With “typical” value:
                                     With lower value,
                                     such that reduction
                                     in peak ~same as
                                     for MPDATA

                        Looks very
   Looks like O(1)      similar to
   donor cell result    MPDATA
   (not shown)
         Rotated Stommel Gyre
• In previous work, fast
  western boundary
  current was aligned
  with principal grid
   – Here it is at 45º, for a
     more discriminating
     test, in Hecht, Wingate
     and Kassis (Ocean
     Modelling, 2000)
            Reference Solution

• Same reference solution (produced more elegantly by Beth
  Evaluation of advection schemes in a
    primitive equation ocean model
• Very simple
   – wind stress
   – heat flux
   – fresh water flux
  with simple flat-bottom
  configuration produce
  idealization of Atlantic
  circulation (Hecht, Bryan
  and Holland, J. Geophys.
  Res., 1998)
    integrated flow
• Vertically integrated flow is as
  predicted by Sverdrup
   – Magnitude of N/S flow proportional
     to curl of wind stress
• Think of northern-most gyre as
  being idealization of “sub-polar
  gyre” (Labrador Sea/Northern
  North Atlantic)
• Think of second northern-most
  gyre as “subtropical gyre” --
  containing the Gulf Stream
         Velocities at level 3

Even though forcing is simple, response is complex

• Atlantic-like
  northern deep
  – This was set
    up through
    asymmetry in
    heat flux
     Gravitational instability and mixing

From a section
through the western
produces 2 dz noise
which creates spurious
MPDATA run without
Some problems require strict limiting

Here, looking at idealized age tracer (“ideal age”), set to 0 at
surface, constant clock source elsewhere. Age shouldn’t be
greater than the duration of the run (or less zero)!
Cost of limiting motivates

               In some simulations CFL for
               passive scalars is
               significantly less restrictive
               than time-step limitation for
               dynamical tracers (density)
               presented by internal waves

               Longer time steps can be
               taken on passive scalars,
               using either instantaneous or
               time-averaged mid-point
 From supercycling of passive scalars to
        “Method of Averaging”

• Supercycling in ocean models hasn’t caught on,
  because of competition from another acceleration
   – This thinking led us into:
• Method of Averaging (MOA)
   – Use of time-averaged (low-pass filtered) advecting
     velocities in momentum equations, to low-pass filter
     wave motions and allow for stable integrations with a
     relatively long time step.
      Does any of this matter?

• “Equatorial nutrient trapping” -- discussed
  for years -- was a consequence of advective
        …and it’s hard to maintain
7 orders of magnitude difference in mixing between
      quasi-horizontal and vertical directions

• Griffies shows how to diagnose spurious,
  numerical mixing
 Now back to our regular programming…

        Special projects in ocean
        modeling with MPDATA
•Method of Averaging (MOA)
  –Use of time-averaged (low-pass filtered) advecting
  velocities in momentum equations, to low-pass filter wave
  motions and allow for stable integrations with a relatively
  long time step.
                Method of Averaging
      DT is CFL-limited time step
      dt is internal-wave limited time step
      DT = M * dt (M is some integer)
1)     First, using O(1) donor cell advection, integrate through DT by
       taking M small steps of size dt.
2)     Then, using a high-order method with the time-averaged advecting
       velocity from 1), repeat integratation through DT in one step.
  Rossby to Kelvin Wave Problem
Problem taken
from Milliff &

Applied with
Method of
Averaging in
Nadiga, Hecht,
Margolin &
               Results with MOA

MOA result, with 8x larger time step, compares well with explicit
reference case
 Kelvin Wave signal at boundary point

Explicit reference
case contains
frequencies, MOA
case is smoother
line (ignore 3rd,
offset curve).
MOA accurately
reproduces low-
pass Kelvin Wave
               Use of MOA
• In forest fire code
  – Reisner et al., Monthly Weather Review, 2000.
      DNS of rotating tank/boundary separation

From Baines                              DNS with
and Hughes, J.
Physical                                 EULAG code
   Time stepping schemes used in ocean models
                                      Momentum equations
                        Scalar eqns
From Griffies 2000                    3D (slow)    2D (fast)

  HIM and POSUM have Forward with PC or FB; other models
  listed only have options for Forward on tracers (scalars).
                   The Future:
          New models under development
          with 2-time-level dynamical cores
•   Bob Higdon’s dy. core uses MPDATA
    Developed as alternative dy. core for
    –   Miami Isopycnic Ocean Model (MICOM)
    –   Hybrid Coordinate Ocean Model (HYCOM)
    MI/HYCOM use form of MPDATA developed
    by Drange & Bleck (‘97) for adv of temperature
•   Los Alamos’ HYPOP using Dukowicz’s
    forward-in-time “remapping”
•   Are finite elements in our future (really)?
Higdon’s dynamical core
        JCP, 2002
Time-evolution of interface
Asymptotic state
               In Summary
• Forward-in-time methods are in common
  use for scalar transport, where problems
  with 2 grid point noise and spurious
  extrema are obvious.
• New models are being developed with fully
  forward-in-time dynamical cores.
  – HOME effort to share code in layered ocean
    models will be largely forward-in-time
TOPEX                         C=1


     Obs      Models
• Gulf Stream, North
  Atlantic Current into the
  “Northwest Corner”, seen
  in Sea Surface Height
              Western boundary currents
                     (boundary jets)

Maltrud and

To top