1. 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

2. 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

3. General circulation models

4. 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.

5. Linear, continuously stratified (LCS) model

6. 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 N b 2 = –g  bz /  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.

7. Now, assume that the vertical mixing coefficients have the special form: ν = κ = A/N b 2 (z). In that case, the last three equations can be rewritten in terms of the operator, (∂ z N b –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.

8. 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, ψ n (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 c 0 =  in which case ψ n (z) = 1 (barotropic mode) or c n is finite so that the integral of ψ n vanishes (baroclinic modes). (2)

9. 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 u n, v n, and p n 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 c n, the Kelvin-wave speed for the mode.

10. Steady-state balances

11. 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 and all mixing terms are not important is defined to be in a state of Sverdrup balance.

12. 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.

13. 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.

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

15. 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

16. Layer models

17. 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 1½-layer model The model allows water to transfer into and out of the layer by means of an across-interface velocity, w 1. In a linear version of the model, h 1 is replaced by H 1, and the model response behaves like a baroclinic mode of the LCS model, and w 1 is then analogous to mixing on density.

19. 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 (w 1 > 0), layer 2 loses the same amount of water, so that mass is conserved. In this case, when h i is replaced by H i the model response separates into two baroclinic modes, similar to the LCS model.

20. Variable-temperature, 2½-layer model If a phenomenon involves upwelling and downwelling by w 1, 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 w 1 causes water parcels to transfer between layers.

21. 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 w i. AAIW NPIW SPLTW thermocline 4½-layer model

22. 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.

24. 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

25. Variable-temperature, 2½-layer model where the density terms are given by Because T i varies horizontally, the pressure gradient depends on z [i.e., p z = –gρ  (  p) z = –g  ρ], within each layer. So, the equations use the depth-averaged pressure gradients in each layer,