Model overview:
A hierarchy of ocean models
A short course on:
Modeling IO processes and phenomena
INCOIS
Hyderabad, India
November 16−27, 2015
References
1) HIGnotes.pdf: Section 2, pages 3−17.
2) InteriorNotes.pdf: Problem 2, pages 19−23.
3) MKM93.pdf: Section 2.
McCreary, J.P., P.K. Kundu, and R.L. Molinari, 1993: A numerical
investigation of dynamics, thermodynamics and mixed-layer processes
in the Indian Ocean. Prog. Oceanogr., 31, 181−244.
4) Vertical modes.pdf: An overview of baroclinic and barotropic
modes.
5) Shankar_notes/Vertical normal modes: An overview of
baroclinic and barotropic modes, with a general discussion of normal
modes.
6) Pressure_1.5Lmodel.pdf: A derivation of the pressure terms in the
1½-layer model.
Introduction
1) General circulation models (GCMs)
2) Linear, continuously stratified
(LCS) model: (barotropic and baroclinic
modes)
3) Steady-state balances
4) Layer ocean models (LOMs)
General circulation models
OGCM equations
A complete set of equations of motion for the ocean has a form
similar to
and provides 7 equations in the 7 unknowns u, v, w, p, ρ, S, and T.
OGCM equations
These equations don’t take into account that density is almost constant
in the ocean.
Adopt the Boussinesq approximation by setting
ρ to a constant in the momentum equations.
This approximation is EXCELLENT. I don’t
know of any situation where it fails.
OGCM equations
These equations don’t take into account that density is almost constant
in the ocean.
Mass conservation can be rewritten
a statement that as you follow a water parcel the only way its
density can change is by expanding or contracting it. Since the
density of sea water is almost constant, Dρ/Dt ≈ 0 and hence
the divergence of v nearly vanishes.
This approximation is EXCELLENT. One impact, however,
is that sound waves are filtered out of the ocean model.
OGCM equations
Finally, many OGCMs adopt the hydrostatic approximation, which
neglects the −γu term and all the w terms in the third equation.
pz = −ρg
This approximation is
usually EXCELLENT.
Linear, continuously
stratified (LCS) model
Simpler ocean models
It is often difficult to isolate basic processes at work in solutions to
such complicated OGCMs. Fortunately, basic processes are illustrated
in simpler systems, providing a language for discussing phenomena
and processes in the more complicated ones. Moreover, OGCM and
solutions to simpler models are often quite similar to each other and to
observations.
Here, we derive the equations for the linear, continuously stratified
(LCS) model, a simpler set of equations that allows for analytic
solutions. It is important to keep in mind the assumptions built into
the simpler equations.
LCS model
momentum
advection
terms.
Their
is studies
sensible
Impose
hydrostatic
relation
by neglecting
wt and
(νw
Drop
thethe
horizontal
Coriolis
term.
I know
of neglect
very
few
that
z) z.
because
thewimpact
linear
terms
known
to
play of
an
important
(often
Dropping
waves
the
order of the
explore
the
ofhigh-frequency
this are
term.
It is certainly
not
important
forVaisala
any
t affects
dominant)
roleofininterest
the
equations.
Nevertheless,
nonlinear
terms
frequency,
not
here. in
Dropping
(νwzSo,
)zthe
filters
out a very
of
the phenomena
considered
this course.
this
assumption
is
are
to belayer
important
manysurface
ocean that
processes
(e.g.,
thinknown
boundary
near thefor
ocean
is dynamically
VERY
GOOD.
instabilities
assumption
QUESTIONABLE,
unimportantand
for eddies).
the rest ofSo,
thethis
flow
field. Thisis assumption
is GOOD.
and can only be assessed by comparing linear solutions carefully with
observations.
LCS
Linearize the equation of
statemodel
to
Then, set κT = κS and combine the T and S equations to obtain a
single density equation. The linearization ignores subtle density
effects in the deep ocean (e.g., caballing) and setting κT = κS deletes
double diffusion. These processes aren’t important for phenomena
considered in this course. So, this assumption is VERY GOOD.
LCS model
Drop
horizontal
ofadensity.
As forthe
themodel
neglect
of thethe
The
derivative
ρbzadvection
is related
fundamental
ocean
frequency,
We
can’t
drop the
wρ
because
it allows
to “know”
z term, to
momentum
advection
this
is zQUESTIONABLE.
Vaisala
frequency,
theterms,
square
of
which
is wρ
that
the ocean
is stratified.
So,
weassumption
linearize
by replacing ρz with
ρbz where ρb(z) is an assumed background density structure of the
ocean. This linearization is common; it was first used by Fjeldstad
Replace
ρbz with
Nb2. is usually SURPRISINGLY GOOD.
(1933). This
assumption
LCS model
Modify the form of vertical diffusion from (κρz)z to (κρ)zz. This
assumption is essential to allow the expansion of solutions into
vertical (barotropic and baroclinic) modes. Since the precise form
of vertical diffusion is not known, it is OKAY.
LCS model
Modify
the form
vertical
from
(κρzlayer.
)z to (κρ)
. This
Wind stress
entersofthe
ocean diffusion
in a surface
mixed
To zzsimulate
assumption
allow
expansion
of as
solutions
this process isinessential
a simple to
way,
wethe
introduce
wind
a “bodyinto
force”
vertical
and Z(z).
baroclinic)
modes.
precise
form
with the(barotropic
vertical profile
The body
forceSince
differsthe
from
an actual
of
vertical
is profile
not known,
it is OKAY.
mixed
layerdiffusion
in that its
is uniform
in space and constant in
time. This representation is CONVENIENT and SENSIBLE.
LCS model
(1)
(2)
(3)
Wind stress enters the ocean in a surface mixed layer. To simulate
this process in a simple way, we introduce wind as a “body force”
Rewrite equations (1) − (3). First, solve (1) for ρ and (2) for w in
with the vertical profile Z(z). The body force differs from an actual
terms pz. Then, insert both expressions into (3).
mixed layer in that its profile is uniform in space and constant in
time. This representation is CONVENIENT and SENSIBLE.
LCS model
Finally,
that(1) − (3). First, solve (1) for ρ and (2) for w in
Rewrite assume
equations
terms pz. Then, insert both expressions into (3).
In which case all the z-operators have the same form, a property
necessary to represent solutions as expansions in vertical modes.
Baroclinic and barotropic modes
Assuming further that the bottom is flat and with boundary
conditions consistent with (2) below, solutions can be represented as
expansions in vertical modes, ψn(z). They satisfy,
(1)
subject to boundary conditions and normalization
(2)
Integrating (1) over the water column gives
(3)
Constraint (3) can be satisfied in two ways. Either c0 =  in which
case ψ0(z) = 1 (barotropic mode) or cn is finite and its value is set
so that the integral of ψn vanishes (baroclinic modes).
Baroclinic and barotropic modes
When N
decreases
depthand
like
ρbb2and
Nb2 arewith
constants
cn is finite (baroclinic modes), the
solutions to (1) are cosine functions,
cos(mz).
andIncorder
solutions
to (1) are
to satisfy
boundary
n is finite,
similar,
except
their
wavelength
conditions
(2), m
must
equal an
increases
and amplitude
decreases
integral number
of half wavelengths
with
in thedepth.
water column, that is,
The values of cn are different from,
but are similar to, those for constant
density.
When cn is infinite, the solution to
(1) that satisfies boundary conditions
(2) is
the barotropic mode of the system.
Mode equations
The solutions for the u, v, and p fields can then be expressed as
where the expansion coefficients are functions of only x, y, and t.
The resulting equations for un, vn, and pn are
Thus, the ocean’s response can be separated into a superposition of
independent responses associated with each mode. They differ only
in the values of cn, the Kelvin-wave speed for each mode.
Equatorial Undercurrent
McCreary (1981a) used the LCS model to study the dynamics of
the Pacific Equatorial Undercurrent (EUC), forcing it by a steady
patch of easterly wind of the separable form
X(x)
The meridional structure Y(y) gradually weakens to zero away from
When the LCS model includes diffusion (A ≠ 0), realistic steady
the equator.
flows can be produced near the equator.
Coastal Undercurrent
McCreary (1981b) obtained
a steady-state, coastal
solution to the LCS model
with damping.
In good agreement with
observations, the solution has
upwelling in the band of wind
forcing, a surface current in
the direction of the wind, and a
subsurface CUC flowing
against the wind.
Comparison of LCS and GCM solutions
The linear model reproduces the GCM solution very well! The
color contours show v and the vectors (v, w).
Steady-state balances
Sverdrup balance
It is useful to extend the concepts of Ekman and Sverdrup
balance to apply to individual baroclinic modes. The complete
equations are
A mode in which the time-derivative terms and all mixing terms are
not important is defined to be in a state of Sverdrup balance.
Ekman balance
It is useful to extend the concepts of Ekman and Sverdrup
balance to apply to individual baroclinic modes. The complete
equations are
A mode in which the time-derivative terms, horizontal mixing
terms, and pressure gradients are not important is defined to be in a
state of Ekman balance.
Layer models
1½-layer model
If a particular phenomenon is surface trapped, it is often useful
to study it with an upper-layer model that focuses on the surface
flow. Such a model is the 1½-layer, reduced-gravity model. Its
equations are
the version
pressure
The
model
allows
water
to transfer
into
and
out ofterms
themode
layer
by
A
linear
ofis
the
model
drops
the
nonlinear
and of
Inwhere
this
case,
the
model
response
behaves
like
a baroclinic
means
ofhmodel,
an
across-interface
w1. Thus, the system can
replaces
the
LCS
where
cn2 = g'velocity,
1 with H
1.
21H1, and w1 is analogous to mixing
allow
for upwelling
downwelling
regions
in the ocean.
on
density.
It is oftenand
useful
to interpret
the response
of the n = 1
so that g' has
a much
smaller
(reduced)
value than g.
baroclinic
mode
as that
of a 1½-layer
model.
2½-layer model
If a phenomenon involves two layers of circulation in the upper
ocean (e.g., a surface coastal current and its undercurrent), then a
2½-layer model may be useful. Without momentum advection,
its equations are
where
= 1,2when
is a layer
andby
theHpressure
gradients
in each
In thisicase,
hi is index,
replaced
the
model
response
separates
i
layer
into
twoare
baroclinic modes, similar to the LCS model.
Variable-density, 1½-layer model
An extended version of the 1½-layer model allows temperature (and
salinity) to change within the layer, a variable-density, 1½-layer
model. Its equations are
Because
T1 varies
horizontally,
pressure-gradient
terms depend
When
It is possible
deep
water
to extend
entrains
the model
intothe
layer
further
1, to
water
allow
with
fortemperature
salinity
to vary
T2
on z since
the
(p)
–gρ.
So,
model
+ (w
+ is in
z = –gρ
mixes
within
into
theplayer.
layer
1Further,
at
rate
itz =can
w be
extended
thethe
positive
to layer
include
part
more
ofthey
wlayers.
), and
1
1
hence T1 cools since T2 < T1.
are replaced by their vertical averages.
1
Variable-density, 4½-layer model
Schematic diagram of the structure of a 4½-layer model used to study
biophysical interactions in the Arabian Sea.
mixed layer
diurnal
thermocline
seasonal
thermocline
main
thermocline
Variable-density, 6½-layer model
Schematic diagram of the structure of a 6½-layer model used to study
the oxygen minimum zones in the Arabian Sea and Bay of Bengal.
mixed layer
diurnal thermocline
seasonal thermocline
main thermocline
upper OMZ
lower OMZ
Why use a variable-density, n½-layer model rather than an OGCM?
Its advantage is its limited vertical resolution: Each layer
corresponds to a well-defined layer or water mass
in thelayer
real ocean.
sub-OMZ
As such, it is computationally very efficient. Its limited vertical
resolution, however, is also a disadvantage, as potentially important
small-vertical-scale processes are filtered out.
Yoshida (2-dimensional) balance
An equatorial balance related to Ekman balance is the 2d, Yoshida
balance, in which x-derivatives are negligible. The equations are.
In this balance, damping is so strong that it eliminates wave
radiation. High-order modes in the McCreary (1981) model of the
EUC are in Yoshida balance.
2-layer model
If the circulation extends to the ocean bottom, a 2-layer model is
useful. Its equations are
for layer 1,
for layer 2, and the pressure gradients are
2-layer model
If the circulation extends to the ocean bottom, a 2-layer model
may be useful. Its equations can be summarized as
where i = 1,2 is a layer index, and the pressure gradients in each
layer are now
In this case, when hi is replaced by Hi the model response
Note that when water entrains into layer 1 (w1 > 0), layer 2
separates into a barotropic mode and one baroclinic mode.
loses the same amount of water, so that mass is conserved.
Variable-temperature, 2-layer model
Because Ti varies horizontally in each layer, the pressure
gradients depend on z (i.e., pz = –gρ  (p)z = –gρ). So, the
equations use the depth-averaged pressure gradients within each
layer,
where the densities are given by
Variable-temperature, 2-layer model
If a phenomenon involves upwelling and downwelling by w1 or
surface heating Q, it is useful to allow temperature (density) to
vary horizontally within each layer.
The 2-layer equations are then
the same equations as for the constant-temperature model except that
the pressure gradients are modified and there are T1 and T2
equations to describe how the layer temperatures vary in time.
Variable-temperature, 2½-layer model
Because Ti varies horizontally, the pressure gradient depends on z
[i.e., pz = –gρ  (p)z = –gρ], within each layer. So, the equations
use the depth-averaged pressure gradients in each layer,
where the density terms are given by
Variable-temperature, 2½-layer model
If a phenomenon involves upwelling and downwelling by w1, it is
useful to allow temperature (density) to vary within each layer.
Equations of motion of are
where the terms
ensure that heat and momentum are conserved when w1 causes
water parcels to transfer between layers.
4½-layer model
Meridional section from a solution to a 4½-layer model of the Pacific
Ocean, illustrating its layer structure across the central basin.
thermocline
SPLTW
NPIW
AAIW
Water can transfer between layers with across-interface velocities wi.