IntroductionIntroduction

Jay McCreary

A short course on:

Modeling IO processes and phenomena

University of TasmaniaHobart, Tasmania

May 4–7, 2009

Focus The focus of the course is on Indian-Ocean dynamics, that is,

understanding the processes that maintain IO circulations and cause their variability on all time scales.

Only a startIt is impossible to cover everything about the IO in this short

time. So, the talks will only provide introductions to various topics of IO dynamics. If you would like to bring up another topic for discussion, please do so.

Discussion Although these talks are called “lectures,” they are meant to be

“discussions.” So, ask questions and bring up issues at any time during the talks.

MistakesI often make mistakes, both in speaking and (likely) in the

slides. Please bring any errors to our attention as soon as you notice them.

TopicsMathematical background (Lectures 1 & 3)

HIG Notes

Mean & annual cycle (Lectures 2a & 2b)SM01, MKM93, SXM09SMJ04, Miyama et al. (2003)

Tsuchiya Jets (Lecture 4)McCreary, Lu, & Yu (2001), Furue et al. (2007, 2009)

Indonesian Throughflow (Lecture 5)PTNE, Godfrey & Weaver (1991)Hirst & Godfrey (1993, 1994)

Intraseasonal variability (Lecture 6)Kessler (2004)

TopicsIO biophysics (Lecture 7)

McCreary et al. (2009) review paper

IO climate (Lecture 8)SXM09 review paper

Supergyre (Wenju Cai)

IO observing systems (Gary Meyers)

New ITF observations (Susan Wijffels)

IO heat variability (Matthew England)

ThanksRichard Coleman (UTas)

Matthew England (UNSW)

Susan Wijffels, Andreas Schiller (CSIRO)

Gary Meyers (IMOS)

Denbeigh Armstrong (UTas)

Model overview:Model overview:A A hierarchy hierarchy of ocean modelsof ocean models

Jay McCreary

A short course on:

Modeling IO processes and phenomena

University of TasmaniaHobart, Tasmania

May 4–7, 2009

ReferencesReferences1) (HIG Notes) McCreary, J.P., 1980: Modeling wind-driven ocean circulation. JIMAR 80-0029, HIG 80-3, Univ. of Hawaii, Honolulu, 64 pp.

These notes contain almost all the mathematics that is discussed in this course and more, the exception being the nonlinear process discussed in Lecture 4 (TJ dynamics). It is not essential to read all the math, but I do hope that some of you may be interested in reading further after the course.

IntroductionIntroduction

1) General circulation models (GCMs)

2) Linear, continuously stratified (LCS) model

3) Layer ocean models (LOMs)

4) Steady-state balances

General circulation models

Equations for a sophisticated OGCM can be summarized in the form

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 simpler solutions are often quite similar to each other and to observations.

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 = –gbz/ is assumed to be a function only of z. Vertical

mixing is retained in the interior ocean. To model the mixed layer, wind stress enters the ocean as a body force with structure Z(z). To expand into vertical normal modes, the structure of vertical mixing of density is modified to (κρ)zz.

Now, assume that the vertical mixing coefficients have the special form: ν = κ = A/Nb

2(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,

subject to boundary conditions and normalization

Integrating (1) over the water column gives

(1)

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).

(2)

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.

The meridional structure Y(y) gradually weakens to zero away from the equator.

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)

Equatorial Undercurrent

When the LCS model includes diffusion (A ≠ 0), realistic steady flows can be produced near the equator.

The linear model reproduces the GCM solution very well! The color contours show v and the vectors (v, w).

Comparison of LCS and GCM solutions

Layer models

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 model. Its equations are

where the pressure is

1½-layer model

The model allows water to transfer into and out of the layer by means of an across-interface velocity, w1.

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.

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

2½-layer model

Note that when water entrains into layer 1 (w1 > 0), layer 2 loses the same amount of water, so that mass is conserved.

In this case, when hi is replaced by Hi the model response separates into two baroclinic modes, similar to the LCS model.

for layer 2, and the pressure gradients are

2-layer model

for layer 1,

If the circulation extends to the ocean bottom, a 2-layer model is useful. Its equations are

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.

Variable-temperature, 2½-layer model

where the density terms are given by

Because Ti varies horizontally, the pressure gradient depends on z [i.e., pz = –gρ (p)z = –gρ], within each layer. So, the equations use the depth-averaged pressure gradients in each layer,

Meridional section from a solution to a 4½-layer model of the Pacific Ocean, illustrating its layer structure across the central basin.

Water can transfer between layers with across-interface velocities wi.

AAIW

NPIWSPLTW

thermocline

4½-layer model

4½-layer model

mixed layer

diurnal thermocline

seasonal thermocline

main thermocline

Schematic diagram of the structure of a 4½-layer model used to study biophysical interactions in the Arabian Sea.

Steady-state balances

It is useful to extend the concepts of Ekman and Sverdrup balance to apply to individual baroclinic modes. The complete equations are

Sverdrup balance

A mode in which the time-derivative terms all mixing terms are not important is defined to be in a state of Sverdrup balance.

It is useful to extend the concepts of Ekman and Sverdrup balance to apply to individual baroclinic modes. The complete equations are

Ekman balance

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.

An equatorial balance related to Ekman balance is the 2d, Yoshida balance, in which x-derivatives are negligible. The equations are.

Yoshida balance

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.