Ocean modelling - Centre of Excellence for Climate System Science

Ocean Modelling:
An Introduction for Everybody
Stephanie Waterman
Climate Change Research Centre
& ARC Centre of Excellence for Climate System Science
University of New South Wales
[www.nasa.gov/topics/earth/features/perpetual-ocean.html]

[www.nasa.gov/topics/earth/features/perpetual-ocean.html]

Ocean Modelling:
An Introduction for Everybody
Stephanie Waterman
Climate Change Research Centre
& ARC Centre of Excellence for Climate System Science
University of New South Wales
[www.nasa.gov/topics/earth/features/perpetual-ocean.html]

What is an ocean model ?
a representation
in the form of
equations / computer code
describing
physical processes
of our understanding of how the ocean works.

What is an ocean model ?
a representation
in the form of
equations / computer code
describing
physical processes
of our understanding of how the ocean works.
•the exchange of energy, mass, and momentum between the ocean
and external sources (e.g. radiation, evaporation, precipitation, river runoff, wind
energy that creates waves or currents etc. etc.)
•ocean movement/dynamics including horizontal advection and vertical convection;
and
•3-dimensional mixing and dissipation processes at scales from molecular to
ocean basin
• ...

There are many types of ocean models...
integration time
number of
processes
Earth Models of
Intermediate Complexity
(EMICs)
Global Climate Models or
General Circulation Models
(GCMs)
conceptual or
process models
detail of description

There are many ways to use ocean models...
• to consider future climate scenarios
• to make operational now-casts and forecasts
• to experimentally investigate hypotheses for
the workings of ocean and climate system
phenomena
• to mechanistically interpret ocean observations
• ...

Ocean vs. Atmospheric Models I
1. liquid vs. gas
• air is a compressible gas; seawater is a
(nearly) incompressible liquid
• this relationship requires a fundamentally
different equation of state:
- atmosphere: ideal gas law (easy!)
- ocean: density = fn(temperature,
salinity,pressure) (hard!)
• BUT (in most applications) we can assume
incompressibility (so water into a box =
water out)

Ocean vs. Atmospheric Models II
2. salinty vs. humidity
• seawater contains dissolved chemicals
known collectively as “salinity”
• ocean models must account for the
effects of salinity on density in an
analogous way that atmospheric
models must account for humidity
[http://en.wikipedia.org/wiki/File:IAPSO_Standard_Seawater.jpg#filelinks]

Ocean vs. Atmospheric Models III
3. vertical structure
• the vertical structure of the ocean and
atmosphere share both similarities and
differences
• both have a well-mixed layer near the surface
where most of the heating and cooling occurs
• BUT mixing occurs (to some degree) throughout
the atmosphere while the ocean is much more
stratified below its thin mixed layer
• because the ocean is so stratified, ocean
models can make assumptions about the
dominance of horizontal processes over those
in the vertical
• ocean modellers can benefit from vertical
coordinate systems that exploit the fact that
much of the action occurs in the near-surface
mixed layer
Temperature Profile: Seafloor to Stratosphere
[source: The COMET program]

Ocean vs. Atmospheric Models IV
4. horizontal structure
• the atmosphere blankets the earth in a
laterally continuous layer; it is pierced by
(relatively) small mountain ranges
• the ocean is bounded on 5 of 6 sides by
complex topography:
- a series of irregularly shaped basins
- fringed by narrow continental shelves
- bottom bathymetry plays on 0(1) role
• there are lateral boundary conditions:
freshwater run-off from the continents
that alters density and currents along the
coast
• the need to resolve both horizontal and
vertical processes along ocean margins
plays a critical role the choice of
horizontal and vertical co-ordinate
systems
[courtesy of Maxim Nikurashin]

Ocean vs. Atmospheric Models V
5. time and length scales of motion
• Steve told you that atmospheric models
are more expensive than ocean models
for a given grid size
• BUT the energy-containing (geostrophic)
scales in the atmosphere are much
larger than those in the ocean
• this means we need to model the ocean
at a finer resolution to resolve the same
“types” of features
• also because the ocean interior is almost
adiabatic (i.e. along-density surfaces)
whereas the atmosphere is (relatively)
well-mixed, the equilibrium timescale of
the ocean is much s l o w e r
• to spin up a ocean model from rest:
1000 + years integration !!!
The space and time scales of various ocean phenomena.
Figure 3. The approximate space and time scales of phenomena of interest that could be investigated from
altimetric measurements of ocean topography with adequate spatial and temporal resolution. The dashed lines
indicate the approximate lower bounds of the space and time scales that can be resolved [source: in SSH fields Chelton constructed (ed) 2001]

Ocean vs. Atmospheric Models
Ocean modellers have it both easy and hard
compared to their atmospheric
counterparts:
• the ocean is (nearly) incompressible:
water in ~ water out
• the ocean is strongly stratified:
horizontal processes dominate over
vertical ones
• no change of state of seawater: just
form ice when T < -1.8 o C
BUT
• domain geometry is complex
• lateral boundary conditions are required
and poorly constrained
• the eddies are smaller
• the spin-up time is longer
• there are fewer observations for
validation
[courtesy of Maxim Nikurashin]

Ocean Model Ingredients I
The$hydrosta,c$equa,ons$
1. the primitive (hydrostatic) equations
du
dt
∇P
+ fk × u = − + F + ν∇
ρ
∂P
∂z
=
− gρ
∂ρ
∂ρ
& ∂w
#
+ u. ∇
H
ρ + w + ρ$
∇
H
. u + ! =
∂t
∂z
% ∂z
"
dT
dt
dS
dt
= κ∇
2
= κ ∇
T
S
+
+
Q T
2
S
Q S
2
u
0
Horizontal$momentum$
Hydrosta,c$balance$
Conserva,on$of$mass$
Conserva,on$of$heat$
Conserva,on$of$salinity$
ρ = ρ( P,
S,
T
)
Nonlinear$equa,on$of$state$
7$coupled$equa,ons$in$7$unknowns:$u,$v,$w,$P,$T,$S,$ρ; all$varying$in$3$spa,al$direc,ons$a
= 7 coupled equations in 7 unknowns: u, v, w, P, T, S,

Ocean Model Ingredients cont.
2. boundary conditions
- basin geometry
- bottom topography
- an atmosphere on top
3. initial conditions (maybe from climatology i.e. the mean
state or a previous already spun up model run)
- initial temperature and salinity fields
- initial velocity fields
4. forcing fields
- shortwave radiation, long-wave radiation, latent heat and sensible heat
at the surface
- evaporation and precipitation at the surface
- land surface run-off at the margins
- winds
- tides

Ocean Model Practicalities I
1. the horizontal grid
Regular Grid
• regular grids consist of
regularly spaced lines
• on a spherical earth CAN’T
have both uniform grid
spacing AND straight lines
• in practice grids tend to be
curvilinear and their internal
spacing tends to vary
• regular lat-lon grids also
have a problem at the poles
where grid lines converge

Ocean Model Practicalities I
1. the horizontal grid
Tri-polar Grid
• a cleaver solution: the
tripolar grid = a circular
grid laid over the Arctic
polar region with two poles
positioned over land
[Schopf 2005 after Murray 1996]

1. the horizontal grid
regular grids:
• are computationally efficient
• have (relatively) straightforward
analysis algorithms
• have benefited from decades of
research experience
BUT
• (for a given latitude) have a
fixed resolution: to increase
resolution near the edge of an
ocean basin (where you want
it!) requires an increase of
resolution everywhere including
out in the middle of the ocean
(where you don’t!)
Ocean Model Practicalities I

1. the horizontal grid
Ocean Model Practicalities I
• irregular grids are designed
to give you more freedom to
put spatial resolution where
you want it
• a common scheme is
composed as a series of
triangles = “finite elements”
• by varying the triangle size
we can construct a nonuniform
horizontal resolution
over the computational
domain
[Gorman et al, 2006]

1. the horizontal grid
irregular grids:
Ocean Model Practicalities I
• are efficient owing to the fact that resolution
can be tailored to need as a function of space
• can accurately represent highly irregular
coastlines and topography (no need for
“staircase” coasts and topography!)
BUT
• have unresolved issues with flows dominated
by geostrophy and advection (characteristics
of large-scale ocean flows!)
• have spatially variable resolution-dependent
physics (e.g. viscosity and diffusivity
coefficients should really be resolution
dependent!)
• have spatially variable spurious diapycnal
mixing associated with the numerical scheme
for advection
(a)
(b)
[http://www.ocean-modeling.org]

1. the horizontal grid
irregular grids:
Ocean Model Practicalities I
• have issues related to the matching of low
resolution regions (say with eddy
parameterizations employed) to highly
resolved regions
• tools to analyze are immature
• are computationally expensive: low-order finite
element models are about an order of
magnitude more expensive than finite
difference models per degree of freedom
(a)
• while common in coastal and estuarine
modelling, irregular grids are uncommon
in large-scale ocean modelling
• CLIVAR’s Working Group on Ocean Model
Development (WGOMD) however anticipate
that a coupled climate model using an
unstructured mesh ocean will emerge
within a decade
(b)
[http://www.ocean-modeling.org]

1. the horizontal grid
Ocean Model Practicalities I
• adaptive grids =
unstructured meshes with a
dynamically changing
resolution responding to the
nature of flow in time
• very cool but very much in
its infancy for ocean
modelling applications
• “there will inevitably come a
point at which the additional
overhead of uniform mesh
refinement where it is not
required is greater than the
overhead associated with the
use of finite elements and
mesh adaptivity” - ICOM
model developer
[Brito-Parada et al, 2012]

2. the vertical grid
Ocean Model Practicalities II
Discre)za)on:*v
• the choice of the vertical co-ordinate system is
loaded because:
- the oceans are forced at the surface; most of
the “action” occurs there
- the oceans are strongly stratified
- the oceans are ~ adiabatic in the interior
Discre)za)on:*ver)cal*coordi
z
Height Terrai
• Simple
• Bounda
• No pressure gradient
errors
resolut
Discre)za)on:*ver)cal*coordinate*
- there is complex bottom bathymetry to deal • Spurious diapycnal
z
σ
with
fluxes
z
• as a consequence there exist a number Height of
Terrain following x
approaches to choose from • Simple
• Boundary layer
• No pressure gradient resolution (BBL)
errors
z
Height
• Spurious diapycnal
σ fluxes
Terrain following
MOM,*POP,*MITgcm,*OPA*
• Pressure gradient
error
ρ
• Spurious diapycnal
fluxes Isopycnal
σ
• Pressu
ρ error
• Spuriou
Iso
fluxes
• Simple
ROMS
• “Exactly
• No reso
unstrati
[courtesy of Sonya Legg]

2. the vertical grid
Ocean Model Practicalities II
1. absolute depth / z-coordinate system
• based on a series of depth levels
• common to add vertical resolution near
the surface by decreasing the spacing
between the levels in the upper ocean
relative to the deep
z
Discre)za)on:*v
Discre)za)o
Discre)za)on:*ver)cal*coordi
Height Terrai
• Simple
• Bounda
• No pressure gradient
errors
resolut
• ADVANTAGES: simple to set up;
computationally efficient; there are no
pressure gradient
Discre)za)on:*ver)cal*coordinate*
errors
• Spurious diapycnal
z
σ
• DISADVANTAGES: increased vertical
fluxes
z
resolution near the lateral boundaries Height z Terrain following x
(i.e. on the continental slopes) requires • Simple
• MOM,*POP,*MITgcm,*OPA*
Boundary layer
the addition of grid cells throughout • No pressure gradient resolution
Height
(BBL)
the basin; spurious diapycnal mixing errors
associated with the numerical advection
• Simple • Pressure gradient
scheme
• Spurious diapycnal
z
σ
error
fluxes
ρ
e.g. MOM (ocean model in ACCESS)
• Spurious diapycnal
Height Terrain following fluxes Isopycnal
σ
• Pressu
ρ error
• Spuriou
Iso
fluxes
• Simple
ROMS
• “Exactly
• No reso
unstrati
•
• No pressure gradient
[courtesy of Sonya Legg]

2. the vertical grid
Ocean Model Practicalities II
Discre)za)on:*v
Discre)za)on:*ver)cal*co
2. “terrain following” / σ-coordinate system
• based on the fractional depth, scaled
from 0 to 1:
- 0.01 level is 1% of the depth of the ocean
- 0.5 level is exactly half the depth of the ocean
- 0.99 level is at 99% of the depth of the ocean
Discre)za)on:*ver)cal*coordi
z
Height Terrai
• Simple
• Bounda
• No pressure gradient
errors
resolut
• ADVANTAGES: mimics the bathymetry
and allows high resolution near the sea
• Pressu
floor regardless
Discre)za)on:*ver)cal*coordinate*
of depth or proximity
• Spurious diapycnal
z
σ
ρ error
to land
fluxes
z
• Spuriou
• DISADVANTAGES: z
pressure gradient Height σ Terrain following x Iso
fluxes
errors; issues with spurious diapycnal • Simple
• MOM,*POP,*MITgcm,*OPA*
Boundary layer • Simple
ROMS
mixing coming from the numerical • No pressure gradient resolution (BBL) • “Exactly
Height
advection scheme
errors
Terrain following • No reso
• Simple
• Pressure gradient unstrati
• Spurious
•
diapycnal
Boundary z
σ
error layer
e.g. ROMS, POM
fluxes
ρ
• No pressure gradient
• Spurious diapycnal
Height
resolution Terrain following fluxes Isopycnal (BBL)
[courtesy of Sonya Legg]
σ
•

2. the vertical grid
Ocean Model Practicalities II
• vertical grid defined by density surfaces
• exploits the fact that below the mixed
layer, ocean currents generally flow
along surfaces of equal density (flow is
“adiabatic”)
z
Discre)za)on:*v
on:*ver)cal*coordinate*
3. density (ρ) / “isopycnal” coordinate
system (“layered models”)
Discre)za)on:*ver)cal*coordi
Height Terrai
• Simple
• Bounda
• No pressure gradient
errors
resolut
• ADVANTAGES: simple, “exactly
Discre)za)on:*ver)cal*coordinate*
isopycnal” (no spurious diapycnal mixing!)
• Spurious diapycnal
z
σ
fluxes
z
• DISADVANTAGES: σ
perform poorly ρ Height Terrain following x
where the ocean is less stratified • (e.g. Simple in
• Boundary layer
shallow water); no resolution in an
• No pressure gradient resolution (BBL)
unstratified Terrain fluid; no following
mixed layer unless Isopycnal
errors
you tack one on; issues with entrainment
• Pressure gradient
• Spurious • diapycnal Simple
z
σ
error
fluxes
ρ
e.g. GOLD, MICOM
resolution (BBL)
• Spurious diapycnal
Height Terrain following fluxes Isopycnal
• Boundary layer
nt
MOM,*POP,*MITgcm,*OPA*
• “Exactly” Adiabatic
z
σ
x
• Pressu
ρ error
• Spuriou
Iso
fluxes
• Simple
ROMS
• “Exactly
• No reso
unstrati
[courtesy of Sonya Legg]

2. the vertical grid
4. hybrid coordinate systems
Ocean Model Practicalities II
• seek to optimize performance by
combining the best suited system in
different regions based on the
dominant processes at work
• HYCOM = z coordinates in the
surface mixed layer and density
coordinates in the stratified ocean
below
• here the vertical coordinates
evolve in both time and space as
the depth of the mixed layer
changes. ADVANTAGE: dynamically
optimized coordinate system gives
improved results; DISADVANTAGE:
high computational cost
[courtesy of Eric Chassingnet]

3. model resolution
Ocean Model Practicalities III
What are we trying to resolve?!?
• like the atmosphere the ocean has
“multiple scale variability” = a
broadband of time and length scales
that are important/on which motions
exhibit variation
• further, processes include coupling
across scales (“non-local interactions”)
and these scale interactions can have
important effects
• the range of important time and space
scales is immense: molecular (mm and
seconds) to basin scale (10 000 km and
1000 years)
• multiple scale variability and scale
interactions make ocean dynamics
fascinating but very challenging to
model!
The space and time scales of various ocean processes.
[adapted from: Chelton (ed) 2001 and Dickey and Chang 2001]

3. model resolution
Ocean Model Practicalities III
• general principles of resolution are the
same for both atmospheric and ocean
models
• there are different rules of thumb: one
is that it takes 5 grid points to
accurately define a feature without
aliasing
• this means 1/8° global resolution with
an average horizontal grid cell of 14
km can accurately depict only features
larger than 56 km
• models with variable grid spacing have
variable resolution - beware of
resolution-dependent physics!
• resolution is not cheap - because of the
CFL* condition, as we shrink the
horizontal grid spacing we must add
vertical layers and decrease the time
step
“every halving of the grid spacing
requires roughly ten times as many
computations”
* no transport faster than one grid cell per time step!

Ocean Model Practicalities III
3. model resolution
• global ocean models often describe their horizontal
resolution with respect to their ability to “permit” or
“resolve” mesoscale (i.e. Rossby radius scale) eddies
1 o “coarse resolution” model
resolution
≥ 1°
~ 0.5°
≤ 0.2°
lingo
“coarse”
“eddy-permitting”
“eddy-resolving”
meaning
no eddies
some eddies
eddies generate at realistic
strength and rate
• “eddy resolving” does NOT mean all eddies are
resolved or that all eddy effects of resolved eddies
are acting !!!
• the spatial resolution of the ocean component of
CMIP5 coupled models is 0.2° to 2°: from “coarse” (no
eddies) to “eddy-permitting” (partially resolved eddy
field)
• the effects of eddies need to be parameterized in
coarse models (more later!); what to do in models that
partially resolve the eddy field is an increasingly
important question
0.1 o “eddy-resolving” model
[courtesy of Peter Gent]

4. parameterizations
Ocean Model Practicalities IV
• processes need to be parameterized in a model
for 2 main reasons:
1. we won’t spend the computational resources
required to directly treat them because they
are either too small or too complex;
2. we don’t understand it well enough to be
represented by an equation
Physical processes in overflows/gravity currents
Final properties, transport
and depth of overflow product
water depend on all these
processes
• processes commonly parameterized in ocean
models include:
- mesoscale eddy effects
- submesoscale eddy effects
- dense overflows
- coastal processes
- surface mixed layer processes
- friction
- sub-grid scale mixing
- ocean-ice interactions
Geostrophic eddies
Neutral
buoyancy
level
• low res models: the main problem is mesoscale
eddies; high res models: submesoscale eddies
(fronts and filaments), internal wave mixing,
details of flow-topography interactions
Entrainment of
ambient water
Shear
instability
Bottom friction
Downslope descent
Upper ocean flow
Isopycnal coordinate models must explicitly include parameterization of entrainment.
Z-coordinate models have difficulties capturing descent without excessive spurious mixing.
All coarse resolution models have difficulties with small-scale topography, e.g. channels.
y
Hydraulic
control
[courtesy of Sonya Legg]
z
x
[courtesy of Bob Weller]
(Goosse*et*al)*
[Goosse et al.]
CICE*model*fundamental*equa)on:*​56/50 =−8.(36)−
6(.,h,0);h=frac)onal*area*covered*by*ice*in*thickness*range*(h,h+
9=rate*of*thermodynamic*ice*growth*
:=redistribu)on*by*ridging*
[courtesy of Max Nikurahsin]

4. parameterizations
Ocean Model Practicalities IV
1. parameterizing mesoscale eddy effects:
• mesoscale eddy effects include:
- mixing along isopycnals
- restratifying / flattening isopycnals
- non-local fluxes: transport by rings, Meddies
- acting as a source of small scale noise/
variability
- modifying air-sea fluxes
- cascading energy to different scales
- ...
Labrador Sea:
Evidence of mesoscale eddies
Labrador Sea
Greenland
• critical eddy parameterizations in
coarse-resolution models are:
1. Along-isopycnal diffusion (Redi)
2. Bolus transport (Gent and
McWilliams or “GM”)
Sea Surface Temperature (10th July 1992)
Sea surface temperature showing the eddy field in the Labrador Sea.
[Jourdain et al. 2008]

1.*Mixing*tracers*along*isop
Ocean
isopycnals.*
Model Practicalities IV
1. Along-isopycnal diffusion (Redi)
= mixing of tracers (temperature,
salinty etc.) along density surfaces
2.Bolus transport (Gent and
McWilliams or “GM”)
A*tracer*is*diffused*along*isopycnals,* & 1+
S
isopycnals.*
1
− S
4. parameterizations − ∇ . u
' τ ' = ∇.
κ K ∇τ
where* K
ρ Redi
=
1. parameterizing mesoscale eddy effects:
$ $$ Redi
+
2 x
1 | S |
% Sx
− ∇. u
' τ ' = ∇.
κ K ∇τ
whe
• mesoscale eddy effects include:
ρ Redi
- mixing along isopycnals
S xσ
x
= −∂ / ∂
end result of
z
- restratifying / flattening isopycnals
mixing by
isopycnal large-scale
- non-local fluxes: transport by rings, Meddies
eddies along
diffusivity (resolved)
- are a source of small scale noise/variability
tracer gradient
density surfaces
- modify air-sea fluxes
& 1 0 S #
- cascade energy to different scales
where
$
x
!
- ...
If*|S|≪1* K = $ 0 1 S !
Redi
y
• critical eddy parameterizations in
$
2 !
coarse-resolution models are:
% S S | S |
x y "
If*|S|≪1* K =
eddy
diffusivity
components of
the (resolved)
isopycnal
slope
(the*components
Re di
[see Redi, 1982]
[courtesy of Sonya Legg]

4. parameterizations
Parameteriza)on
Ocean Model Practicalities IV
1. parameterizing mesoscale eddy effects:
• mesoscale eddy effects include:
- mixing along isopycnals
- restratifying / flattening isopycnals
- non-local fluxes: transport by rings, Meddies
- are a source of small scale noise/variability
- modify air-sea fluxes
- cascade energy to different scales
- ...
• critical eddy parameterizations in
coarse-resolution models are:
1. Along-isopycnal diffusion (Redi)
2.Bolus transport (Gent and
McWilliams or “GM”)
= represents the advective or transport effect
of eddies by means of a “bolus” velocity
2.*Bolus*transport*(Gent*and
2.*Bolus*transport*(Gent*an
− ∇. u'
' τ ''
=
−∇.
. τ u
Parameteriza)ons*of*mesoscale
u * & − ∂ z(
κ GMS
x)
#
x
− ∇. u'
τ ' = −∇.
τ u *
“bolus velocity”
Where*​3 ↑∗ *is*the*bolus*velocity*
& u *#
& − ∂
z
( κGM
Sx
) #
. $
τ
!
u
$
!
u*
= $ v*
! =
* $
= − ∇ ..
∂
z
( κGM
S
y
) !
$ ! $
!
% w*
" % ∂
x
( κGM
Sx
) + ∂
y
( κGM
S
y
)"
− ∇ . τ u * = ∇ .
( )
κ ∇ τ
K GM GM
*
Where*​3
Where*​3 effect of eddies
↑∗
↑∗
*is*the*bolus*velocity*
*is*the*bolus*velocity*
& u *
# & − ∂ ( κ S ) #
z GM x
$
!
$
!
!
u
*
=
$
v
*
!
=
$
− ∂
z
z
((
κGM
GM
S
y
y
))
!!
2.*Bolus*transport*(Gent*and*McWilliams)**
$
!
$
!!
%
w
*
"
%
∂
x(
(
κGM
GM
Sx
x
))
+ ∂
y
y
((
κGM
GM
SS
y
y
))
""
−
∇
- mixes “isopycnal thickness” to flatten density
surfaces without mixing density (to mimic baroclinic instability)
. u τ τ u
end result of mixing
by the “advective”
( ))
components of
κ the (resolved)
∇
τ
GM K GM
K GM GM
isopycnal slope
“GM
eddy
diffusivity”
& 0 0 − Sx
#
$
!
KGM
= $ 0 0 − S
y !
$
!
% Sx
S
y
0 "
An*an)Osymmetric*tensor*
K
GM
G
z
κ = αL | S | N
2
κ GM =eddy*coefficient$ S
GM
κ GM 2 [see (Visbeck*et*al,*1994)*
GM
κ GM =eddy*coefficient$ κ
Gent and 2
GM =
McWilliams, αL 1990]
| | S
(A
Tend
isopy
dens
An A
L |
[courtesy of Sonya Legg]

4. parameterizations
Ocean Model Practicalities IV
1. parameterizing mesoscale eddy effects:
• adding along-isopycnal diffusivity and bolus transport “fixed” the problem of widespread
convection and localized it to the few deep water formation sites as observed
constant horizontal diffusivity
with along-isopycnal diffusivity and
bolus transport
Locations of deep convection/deep water formation in a 3 o x 3 o global model.
[JDanabasoglu et al.1994]
[courtesy of Sonya Legg]

4. parameterizations
Ocean Model Practicalities IV
1. parameterizing mesoscale eddy effects:
• adding along-isopycnal diffusivity and bolus transport “fixed” the problem of widespread
convection and localized it to the few deep water formation sites as observed
There once was an ocean model called MOM,
constant horizontal diffusivity
with along-isopycnal diffusivity and
bolus transport
which occasionally used to bomb,
but eddy advection, and much less convection,
turned it into a stable NCOM*.
* Navy Coastal Ocean Model.
Locations of deep convection/deep water formation in a 3 o x 3 o global model.
[JDanabasoglu et al.1994]
[courtesy of Sonya Legg]

4. parameterizations
2. vertical mixing schemes
Ocean Model Practicalities IV
Mixed*layer*
Eddy*diffusivity*
w ' τ ' ( d)
w ' τ ' ( d)
Mixed*layer*
ver)cal*fluxes*
Eddy*diffusivity*
& ∂
τ $ #
$
% ∂z
= −κ
!
τ
−γ
τ
DownOgradient*
% ∂z
"
1994)*
& ∂ τ #
= −κ
$ −γ
!
τ
"
end
diffusion*
• when the ocean becomes statically ver)cal*fluxes* result of eddy
DownOgradient*
downgradient
vertical
non local
Nondimensional*
unstable ( z > 0 ) vertical over-turning
diffusivity
KOprofile*parameteriza)on*(Large*et*al,*
mixing by diffusion* shape*func)on*
flux
diffusion
should occur but cannot because we
eddies 1994)*
Eddy*diffusivity*
make the hydrostatic approximation
Nondimensional*
& d # & d # & d #
κ $ ! = hw $ ! G$
!
(vertical acceleration is excluded!)
& ∂ τ # τ
​1↓2 τis*nonOzero*only*for*
w ' τ ' ( d)
= −κ
$ ! shape*func)on*
% h " % h " % h "
τ
−γ
τ
tracers*in*convec)ve*forcing*
• so vertical mixing must be accomplished % ∂z
Mixed*layer*
"
condi)ons,*to*account*for*
mixed Mixed*layer*
vertical
Ver)cal*
via a very large coefficient of
ver)cal*fluxes*
counterOgradient*flux$
vertical
non-dimensional
DownOgradient*
& d #
depth*
& d #
velocity
& dvelocity*scale*
#
diffusion*
diffusion
κ $ ! =
Nonlocal*flux*
shape function
hw $ ! Gscale
τ
$ !
τ
• many models use the K-Profile % h "
Allows*for*gradients*in*mixed*layer,*and*
% h " % h
Nondimensional*
Convec)ve*buoyancy*flux*
"
shape*func)on* penetra)ve*convec)on**
Parameterization (Large et al., 1994)
Mixed*layer*
= large mixing in the upper ocean due & dto # many & d # & d # Ver)cal*
κ τ $ ! = hwτ
$ ! G
processes (but dominated by wind) and very depth*
$ !
% h " % h " % h " z* velocity*scale* h*
much weaker mixing in the deep ocean Mixed*layer* due to Ver)cal*
Convec)vely*
internal wave breaking and tides
depth* velocity*scale*
Allows*for*gradients*in*mixed*layer,*and*
modified*
buoyancy*
penetra)ve*convec)on** b*
Allows*for*gradients*in*mixed*layer,*and*
penetra)ve*convec)on**
*
1994)*
Nonlocal*flux*
Nonlocal*flux*
Conve
z*
z*
Con

Sources of Error
Errors in ocean model arise from several
sources:
• mathematical error from the truncation
and rounding of numbers from one time
step to the next
• propagating errors from errors in the
forcing fields
• inaccurate parameterizations
• missing processes or missing interactions
between processes
• spurious (numerical) diapycnal mixing
associated with the numerical advection
scheme

Validation
• validation is a challenge because ocean observations are sparse
• things are tough but getting better. Now:
- satellite measurements of sea surface height (SSH), sea surface temperature (SST), salinity
and winds
- in situ measurements from coastal stations, buoys, fixed current meters, ships, drifters, floats
and gliders
• sustained observations are critical to ocean model development !!!

That’s nice, but what is needed from the ocean
component model to get climate change correct?
• to get heat and CO 2 uptake correct
• a good representation of the circulation including the meridional
overturning circulation (MOC)
• a good vertical mixing scheme to get correct mixed layer depths and
deep water formation in small, local regions as observed
• a good parameterization of the effects of mesoscale eddies - they
aren’t resolved in (most) climate models
• ...
[courtesy of Peter Gent]

your challenges to solve:
Ocean Models: The Future
“The ideal ocean climate model has high enough resolution to resolve eddies and
topography, zero numerical diffusion and is efficient enough to integrate for 1000s
of years.”
• model biases and model drift
• projections of sea-level rise: most current GCMs are Boussinesq and must calculate
the steric contribution to sea-level rise a-posteriori
• the realistic representation of mixing: how much, where? why?
• spurious mixing: models are too diffusive; in z-coordinate resolution models numerical
mixing depends on resolution
• overflows
• getting the energy out of the mesoscale eddy field
• sub-mesoscale effects/parameterization
• mixed layer depth and dynamics
• effects of the internal wave field
• parameterizations for partially resolved eddy fields; resolution-dependent
parameterizations
• ICE: dynamic ice-sheets and ice shelves, iceberg transport (lack of leads to cold-fresh
bias around Antarctica!)
• ...

Want to know more?
• MIT Open Courseware: “12.984 Atmospheric and Oceanic Modeling”
http://ocw.mit.edu/courses/earth-atmospheric-and-planetary-sciences/12-950-
atmospheric-and-oceanic-modeling-spring-2004/index.htm
• Griffies, S M, 2004: Fundamentals of Ocean Climate Models, Princeton, NJ:
Princeton University Press, 518 pp.
• Griffies, S M, C Böning, F O Bryan, E P Chassignet, R Gerdes, H Hasumi, A C
Hirst, A M Treguier, and D Webb, 2000: Developments in ocean climate
modelling. Ocean Modelling, 2, 123-192.
• Griffies, S M, and A Adcroft, 2008: “Formulating the equations of ocean
models” In Ocean Modeling in an Eddying Regime, Geophysical Monograph 177,
M. W. Hecht, and H. Hasumi, eds., Washington, DC, American Geophysical Union,
281-318.
• Griffies, S M, 2009: “Science of ocean climate models” In Encyclopedia of Ocean
Sciences, 2nd edition, Elsevier, DOI: 10.1016/B978-012374473-9.00714-1.
• Griffies, S M, A Adcroft, A Gnanadesikan, R W Hallberg, M J Harrison, S Legg, C
M Little, M Nikurashin, A Pirani, B L Samuels, J R Toggweiler, and G K Vallis, et
al., 2010: Problems and prospects in large-scale ocean circulation models In
Ocean Obs '09, 21-25 September, Venice, Italy, ESA Special Publication, DOI:
10.5270/OceanObs09.cwp.38.

[courtesy of Andreas Klocker, CoE]