Model overview:A hierarchy of ocean models

A short course on:

Modeling IO processes and phenomena

INCOISHyderabad, India

November 16−27, 2015

References1) 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

A complete set of equations of motion for the ocean has a form similar to

OGCM equations

and provides 7 equations in the 7 unknowns u, v, w, p, ρ, S, and T.

These equations don’t take into account that density is almost constant in the ocean.

OGCM equations

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.

These equations don’t take into account that density is almost constant in the ocean.

OGCM equations

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.

Finally, many OGCMs adopt the hydrostatic approximation, which neglects the −γu term and all the w terms in the third equation.

OGCM equations

pz = −ρgThis 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

Drop the momentum advection terms. Their neglect is sensible because the linear terms are known to play an important (often dominant) role in the equations. Nevertheless, the nonlinear terms are known to be important for many ocean processes (e.g., instabilities and eddies). So, this assumption is QUESTIONABLE, and can only be assessed by comparing linear solutions carefully with observations.

Drop the horizontal Coriolis term. I know of very few studies that explore the impact of this term. It is certainly not important for any of the phenomena considered in this course. So, this assumption is VERY GOOD.

Impose the hydrostatic relation by neglecting wt and (νwz)z. Dropping wt affects high-frequency waves of the order of the Vaisala frequency, not of interest here. Dropping (νwz)z filters out a very thin boundary layer near the ocean surface that is dynamically unimportant for the rest of the flow field. This assumption is GOOD.

LCS modelLinearize the equation of state 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 advection of density. As for the neglect of the momentum advection terms, this assumption is QUESTIONABLE.

We can’t drop the wρz term, because it allows the model to “know” that the ocean is stratified. So, we linearize wρz 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 (1933). This assumption is usually SURPRISINGLY GOOD.

The derivative ρbz is related to a fundamental ocean frequency, the Vaisala frequency, the square of which is

Replace ρbz with Nb2.

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 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.

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” with the vertical profile Z(z). The body force differs from an actual mixed layer in that its profile is uniform in space and constant in time. This representation is CONVENIENT and SENSIBLE.

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” with the vertical profile Z(z). The body force differs from an actual mixed layer in that its profile is uniform in space and constant in time. This representation is CONVENIENT and SENSIBLE.

LCS model

Rewrite equations (1) − (3). First, solve (1) for ρ and (2) for w in terms pz. Then, insert both expressions into (3).

(1)

(2)

(3)

LCS model

Rewrite equations (1) − (3). First, solve (1) for ρ and (2) for w in terms pz. Then, insert both expressions into (3).

Finally, assume that

In which case all the z-operators have the same form, a property necessary to represent solutions as expansions in vertical 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,

subject to boundary conditions and normalization

Integrating (1) over the water column gives

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).

(3)

(1)

Baroclinic and barotropic modes

(2)

Baroclinic and barotropic modes

When ρb and Nb2 are constants and

cn is finite (baroclinic modes), the solutions to (1) are cosine functions, cos(mz).

In order to satisfy boundary conditions (2), m must equal an integral number of half wavelengths in the water column, that is,

When Nb2 decreases with depth like

and cn is finite, solutions to (1) are similar, except their wavelength increases and amplitude decreases with depth.

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.

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.

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.

Mode equations

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

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)

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

Equatorial Undercurrent

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 (1981b) obtained a steady-state, coastal solution to the LCS model with damping.

Coastal Undercurrent

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

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

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.

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, horizontal mixing terms, and pressure gradients are not important is defined to be in a state of Ekman balance.

Ekman balance

Layer models

where the pressure is

so that g' has a much smaller (reduced) value than g.

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

1½-layer model

The model allows water to transfer into and out of the layer by means of an across-interface velocity, w1. Thus, the system can allow for upwelling and downwelling regions in the ocean.

A linear version of the model drops the nonlinear terms and replaces h1 with H1.In this case, the model response behaves like a baroclinic mode of the LCS model, where cn

2 = g'21H1, and w1 is analogous to mixing on density. It is often useful to interpret the response of the n = 1 baroclinic mode as that of a 1½-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 i = 1,2 is a layer index, and the pressure gradients in each layer are

2½-layer model

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

Variable-density, 1½-layer modelAn 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, the pressure-gradient terms depend on z since pz = –gρ (p)z = –gρ. So, in the layer model they

are replaced by their vertical averages.

When deep water entrains into layer 1, water with temperature T2 mixes into layer 1 at the rate w1

+ (w1+ is the positive part of w1), and

hence T1 cools since T2 < T1.

It is possible to extend the model further to allow for salinity to vary within the layer. Further, it can be extended to include more layers.

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.

Variable-density, 4½-layer model

mixed layerdiurnal thermoclineseasonal thermocline

main thermocline upper OMZ

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.

Variable-density, 6½-layer model

lower OMZ

sub-OMZ layer

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 the real ocean. 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.

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

Yoshida (2-dimensional) 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.

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

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

2-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 a barotropic mode and one baroclinic mode.

Variable-temperature, 2-layer model

where the densities are given by

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

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

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,

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.

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