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Introduction General circulation models (GCMs) Linear, continuously stratified (LCS) model: (barotropic and baroclinic modes) Layer ocean models (LOMs) The LCS model introduces the concepts of barotropic and baroclinic modes, which have proven to be an important tool for describing and understanding ocean dynamics. It is useful to extend the concepts of Ekman and Sverdrup balances to individual baroclinic modes.
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A hierarchy of ocean models
Model overview: A hierarchy of ocean models Jay McCreary Jay
McCreary A mini-course on: Large-scale Coastal Dynamics University
of Tasmania Hobart, Australia March, 2011 In addition to
observations, ocean models are the tools that we use to understand
ocean phenomena.They range in dynamical sophistication from simple,
1-layer systems to state-of-the-art OGCMs.Solutions to them are
obtained both analytically (paper and pencil) and numerically (with
computers). Which of these models and approaches you use depends on
the purpose of your particular project: If you are trying to
understand basic physics (then use a simple model) or to simulate
reality (then use an OGCM). Generally, it is not possible to
understand a particular phenomena completely without using a
hierarchy of models, that is, to obtain solutions to a suite of
models that vary from simple to complex. Introduction General
circulation models (GCMs)
Linear, continuously stratified(LCS) model: (barotropic and
baroclinicmodes) Layer ocean models (LOMs) The LCS model introduces
the concepts of barotropic and baroclinic modes, which have proven
to be an important tool for describing and understanding ocean
dynamics. It is useful to extend the concepts of Ekman and Sverdrup
balances to individual baroclinic modes. General circulation models
Equations for a sophisticated OGCM can be summarized in the
form
Note that usually the density equation is split into two parts: one
for temperature and another for salinity. It is not possible to
obtain analytic solutions to such a complicated model.So, if (under
some circumstances) difficult terms (like nonlinearities) are
small, then it is sensible to look at solutions to models that
neglect them. In the third equation, pz should be pz/oand g should
be g/o. It is often difficult to isolate basic processes at work in
solutions to such complicated sets of equations.Fortunately, basic
processes are illustrated in simpler systems, providing a language
for discussing phenomena and processes in more complicated ones.
Moreover, GCM and solutions to simpler models are often quite
similar to each other and to observations. Linear, continuously
stratified (LCS) model
The advantage of the LCS model is that solutions can be represented
as expansions in vertical modes.Indeed, the concept of vertical
modes emerged from this model. Although not strictly valid in more
complex systems, modes are often used to understand the flows that
develop in them. Virtually all the advances in equatorial dynamics
emerged from soliutions to this type of 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 To model the mixed layer,
wind stress enters the ocean as a body force with structure Z(z).
Note how the equations differ from those of the OGCM (toggle from
one slide to the other): There are no momentum advection terms, the
hydrostatic approximation is assumed, the horizontal
density-advection terms are dropped, and the wz term is linearized
by fixing Nb to be an externally prescribed function of z alone. To
expand into vertical normal modes, the structure of vertical mixing
of density is modified to ()zz. where Nb2 = gbz/ is assumed to be a
function only of z. Vertical mixing is retained in the interior
ocean. Now, assume that the vertical mixing coefficients have the
special form: = = A/Nb2(z).In that case, the last three equations
can be rewritten in terms of the operator, (zNb2z), as follows
Toggle from the previous slide to this one, and explain how the
last three equations were rewritten. 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, n(z).They satisfy, (1) subject to boundary
conditions and normalization Integrating (1) over the water column
gives Integrate the eigenfunction equation from D to 0.Because of
the boundary conditions, the left hand side is identically zero,
which implies that either cn or that n integrates to zero. These
eigenfunctions are the famous baroclinic and barotropic modes of
the ocean.In many papers, (1) will be used to determine the
vertical structure of the vertical modes of the ocean. Modes are
often invoked in situations where they are not strictly valid
(because of nonlinearities or non-flat bottom topography), such as
for analyzing observations in the real world and in OGCMs. (2)
Constraint (2) can be satisfied in two ways.Either c0 = in which
case n(z) = 1 (barotropic mode) or cn is finite so that the
integral of n 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 1) One advantage of
this simplification is that you can find solutions analytically. 2)
Another is that the cost of obtaining such a solution numerically
is much reduced over that of an OGCM. 3) But have you thrown out
essential physics?In the northern and tropical Indian Ocean, which
is dominated by the propagation of (nearly) linear baroclinic
waves, the answer generally appears to be a qualified no. Thus, the
oceans 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 the mode. Steady
response to switched-on y
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. McCreary
(1981) obtained a steady-state, coastal solution to the LCS model
with damping. There is a surface coastal jet in the direction of
the wind, and also an oppositely directed Coastal Undercurrent.
Comparison of LCS and GCM solutions
The two solutions are very similar, showing that the flows are
predominantly linear phenomena. Differences are traceable to the
advection of density in the GCM. The linear model reproduces the
GCM solution very well!The color contours show v and the vectors
(v, w). 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 1-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
1-layer, reduced-gravity model.Its equations are In a linear
version of the model, h1 is replaced by H1, and the model response
behaves like a baroclinic mode of the LCS model, and w1 is then
analogous to mixing on density. The model allows water to transfer
into and out of the layer by means of an across-interface velocity,
w1. where the pressure is The layer is the deep ocean, assumed to
be so deep that it is essentially quiescent. 2-layer model If the
circulation extends to the ocean bottom, a 2-layer model may be
useful.Its equations can be summarized as In this case, when hi is
replaced by Hi the model response separates into a barotropic mode
and one baroclinic mode. Note that when water entrains into layer 1
(w1 > 0), layer 2 loses the same amount of water, so that mass
is conserved. where i = 1,2 is a layer index, and the pressure
gradients in each layer are now 2-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 2-layer model
may be useful.Its equations can be summarized as In this case, when
hi is replaced by Hi the model response separates into two
baroclinic modes, similar to the LCS model. where i = 1,2 is a
layer index, and the pressure gradients in each layer are
Variable-temperature, 2-layer model
If a phenomenon involves upwelling and downwelling by w1 orsurface
heating Q, it is useful to allow temperature (density) to vary
horizontally within each layer. The 2-layer equations are then More
complex layer models can be devised.In this variable-temperature,
2-layer model, temperature varies in each of the layers, and heat
and momentum are conserved when water particles transfer between
layers. 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, 2-layer model
Because Ti varies horizontally in each layer, the pressure
gradients depend on z (i.e.,pz = g(p)z = g).So, the equations use
the depth-averaged pressure gradients within each layer, There is a
derivation of the depth-averaged pressure gradients when 2 is
constant in ThermalForcing.pdf. where the densities are given
by