Model overview:
A hierarchy of ocean models
Jay McCreary
Summer School on:
Dynamics of the North Indian Ocean
National Institute of Oceanography
Dona Paula, Goa
June 17 – July 29, 2010
Introduction
1) General circulation models (GCMs)
2) Linear, continuously stratified
(LCS) model: (barotropic and baroclinic
modes)
3) Layer ocean models (LOMs)
4) Steady-state balances
General circulation models
Linear, continuously
stratified (LCS) model
Equations: A useful set of simpler equations is a version of the GCM
equations linearized about a stably stratified background state of no
motion. (See the HIG Notes for a discussion of the approximations
involved.) The resulting equations are
where Nb2 = –gbz/ is assumed to be a function only of z. Vertical
mixing is retained in the interior ocean.
To
theinto
mixed
layer,normal
wind stress
enters
the ocean
a body
Tomodel
expand
vertical
modes,
the structure
ofas
vertical
force
with
mixing
of structure
density isZ(z).
modified to (κρ)zz.
Now, assume that the vertical mixing coefficients have the special
form: ν = κ = A/Nb2(z). In that case, the last three equations can be
rewritten in terms of the operator, (∂zNb–2∂z), as follows
Since the z operators all have the same form, under suitable
conditions (noted next) we can obtain solutions as expansions in the
eigenfunctions of the operator.
Vertical modes: Assuming further that the bottom is flat and with
boundary conditions consistent with those below, solutions can be
represented as expansions in the vertical normal (barotropic and
baroclinic) modes, ψn(z). They satisfy,
(1)
subject to boundary conditions and normalization
Integrating (1) over the water column gives
(2)
Constraint (2) can be satisfied in two ways. Either c0 =  in which
case ψn(z) = 1 (barotropic mode) or cn is finite so that the integral
of ψn vanishes (baroclinic modes).
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 value of cn, the Kelvin-wave speed for the mode.
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.
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.
Equatorial Undercurrent
McCreary (1981) 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.
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).
Layer models
1½-layer model
If a particular phenomenon is surface trapped, it is often useful to
study it with a model that focuses on the surface flow. Such a model
is the 1½-layer, reduced-gravity model. Its equations are
where the pressure is
In
a linear
the model,
h1 is into
replaced
by H
andlayer
the model
The
modelversion
allows of
water
to transfer
and out
of1,the
by
response
like a baroclinic
modewof
the LCS model, and w1 is
means ofbehaves
an across-interface
velocity,
1.
then analogous to mixing on density.
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. Its equations can be summarized as
where i = 1,2 is a layer index, and the pressure gradients in each layer
are
Note
entrainsby
into
1 (w1response
> 0), layer
2 losesinto
In
thisthat
case,when
whenwater
hi is replaced
Hi layer
the model
separates
the same
amount
of water,
so to
that
conserved.
two
baroclinic
modes,
similar
themass
LCSismodel.
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.
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
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
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