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, 1½-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 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.

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/ρo and –ρg should be –ρg/ρo. 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 wρz 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 (κρ)zz. where Nb2 = –gbz/ 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: 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 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, ψn(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 ψn 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 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 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 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 the mode.

Steady response to switched-on τy 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

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 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 “½” 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

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 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 or surface 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, 2½-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 = –gρ  (p)z = –gρ). So, the equations use the depth-averaged pressure gradients within each layer, There is a derivation of the depth-averaged pressure gradients when ρ2 is constant in ThermalForcing.pdf. where the densities are given by