Forward-in-time Methods in Ocean Modeling

Matthew HechtLos Alamos National Lab

Primitive Equation Ocean Models

• 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– implying circulation must be very nearly correct

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 tracerthrough thewestern boundarycurrent once.

Reference solutionis producednumerically.

Selected resultsAs on previous slide Centered leapfrog

O(3) upwindWith 1antidiffusivepass

Centered cases with diffusionWith “typical” value:

With lower value,such that reductionin peak ~same asfor MPDATA

Looks verysimilar toMPDATA

Looks like O(1)donor cell result(not shown)

Rotated Stommel Gyre

• In previous work, fast western boundary current was aligned with principal grid axis– 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 Wingate).

Results

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)

Vertically-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

Poleward transports

• Atlantic-like northern deep water formation– This was set

up through asymmetry in heat flux forcing

Gravitational instability and mixing

From a section through the western boundary.

Centered-leapfrog produces 2 dz noise which creates spurious instabilities.

MPDATA run without limiting

Some problems require strict limiting

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

Cost of limiting motivates “supercycling”

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 velocity.

From supercycling of passive scalars to “Method of Averaging”

• Supercycling in ocean models hasn’t caught on, because of competition from another acceleration technique– 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 error

…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

Special projects in ocean modeling with MPDATA

Now back to our regular programming…

•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

IfDT is CFL-limited time step

anddt is internal-wave limited time step

whereDT = 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 ProblemProblem taken from Milliff & McWilliams (1994)

Applied with Method of Averaging in Nadiga, Hecht, Margolin & Smolarkiewicz (1997).

Results with MOA

MOA result, with 8x larger time step, compares well with explicitreference case

Kelvin Wave signal at boundary point

Explicit reference case contains higher frequencies, MOA case is smoother line (ignore 3rd, offset curve).

MOA accurately reproduces low-pass Kelvin Wave signal.

Use of MOA

• In forest fire code– Reisner et al., Monthly Weather Review, 2000.

DNS of rotating tank/boundary separation

From Baines and Hughes, J. Physical Oceanography, 1996.

DNS with EULAG code

Time stepping schemes used in ocean modelsMomentum equations

3D (slow) 2D (fast)Scalar eqns

HIM and POSUM have Forward with PC or FB; other modelslisted only have options for Forward on tracers (scalars).

From Griffies 2000

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 coreJCP, 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

• Gulf Stream, North Atlantic Current into the “Northwest Corner”, seen in Sea Surface Height

C=1

C=1/2

C=1/4

TOPEX

Obs Models

Western boundary currents (boundary jets)

Maltrud and McClean,

Ocean Modelling,

2004