1. A hierarchy of ocean models
Model overview:
A hierarchy of ocean models
A short course on:
Modeling IO processes and phenomena
INCOIS
Hyderabad, India
November 16−27, 2015
Comment on the use of a model hierarchy. Simpler solutions provide the building blocks by which we can understand the complex solutions to OGCMs. Solutions to simpler models often compare well with observations, as well as solutions to OGCMs, and it is usually easier to identify the key dynamical processes at work in them.
Solutions to simpler models provide a dynamical “language” (e.g., Rossby, Kelvin, and gravity waves) for discussing solutions to more complex ones.

2. References 1) 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.

3. Introduction General circulation models (GCMs)
Linear, continuously stratified (LCS) model: (barotropic and baroclinic modes)
Steady-state balances
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.

4. General circulation models

5. and provides 7 equations in the 7 unknowns u, v, w, p, ρ, S, and T.
OGCM equations
A complete set of equations of motion for the ocean has a form similar to
and provides 7 equations in the 7 unknowns u, v, w, p, ρ, S, and T.

6. OGCM equations
These equations don’t take into account that density is almost constant in the ocean.
A nice discussion of the incompressibility of sea water can be found at:
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.
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.

7. OGCM equations
These equations don’t take into account that density is almost constant in the ocean.
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.
The quantity, Dq/Dt, is the “material derivative” of q. It says how property q of a water parcel changes in time as you follow the parcel.
Often OGCMs assume the ocean is hydrostatic, that is, all the w-terms in the third equation are dropped.
Further, they neglect the horizontal Coriolis force term, −γu.
A nice discussion of the incompressibility of sea water can be found at:

8. OGCM equations
Finally, many OGCMs adopt the hydrostatic approximation, which neglects the −γu term and all the w terms in the third equation.
This approximation is usually EXCELLENT.
pz = −ρg
The quantity, Dq/Dt, is the “material derivative” of q. It says how property q of a water parcel changes in time as you follow the parcel.
Often OGCMs assume the ocean is hydrostatic, that is, all the w-terms in the third equation are dropped.
Further, they neglect the horizontal Coriolis force term, −γu.
A nice discussion of the incompressibility of sea water can be found at:

9. 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.

10. 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.

11. LCS model
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.
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.

12. LCS model Linearize 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.

13. LCS model
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.
Drop horizontal advection of density. As for the neglect of the momentum advection terms, this assumption is QUESTIONABLE.

14. 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.

15. 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.

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

17. LCS model
Note that the first three equations form a set of 3 equations in the 3 unknowns, u, v, and p. Once they are solved, w and ρ are then known in terms of p.
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.
Rewrite equations (1) − (3). First, solve (1) for ρ and (2) for w in terms pz. Then, insert both expressions into (3).

18. Baroclinic and barotropic 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,
(1)
subject to boundary conditions and normalization
(2)
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.
(3)
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).

19. Baroclinic and barotropic modes
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 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,
The depth scale b = 1000 m.
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.
When cn is infinite, the solution to (1) that satisfies boundary conditions (2) is
the barotropic mode of the system.

20. Mode equations
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 each mode.

21. Equatorial Undercurrent
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
When the LCS model includes diffusion (A ≠ 0), realistic steady flows can be produced near the equator.
X(x)
There is a big difference between the response to low-order and high-order modes, the former being in Sverdrup balance and the latter in “Yoshida” balance. Zonal flow in the EUC is associated with intermediate modes.
The meridional structure Y(y) gradually weakens to zero away from the equator.

22. Coastal 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.
There is a surface coastal jet in the direction of the wind, and also an oppositely directed Coastal Undercurrent.

23. 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).

24. Steady-state balances

25. 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.

26. 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.

27. Layer models

28. 1½-layer model
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
The “½” layer is the deep ocean, assumed to be so deep that it is essentially quiescent.
A linear version of the model drops the nonlinear terms and replaces h1 with H1.
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.
where the pressure is
so that g' has a much smaller (reduced) value than g.
In this case, the model response behaves like a baroclinic mode of the LCS model, where cn2 = 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.

29. 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. Without momentum advection, its equations are
where i = 1,2 is a layer index, and the pressure gradients in each layer are
In this case, when hi is replaced by Hi the model response separates into two baroclinic modes, similar to the LCS model.

30. Variable-density, 1½-layer model
An 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
A variable-density, 1½-layer model is useful, for example, because it changes the temperature (and salinity) within the layer. Such a model, for example, simulates the response to upwelling of deep water into layer 1 more realistically.
Because T1 varies horizontally, the pressure-gradient terms depend on z since pz = –gρ  (p)z = –gρ. 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.

31. Variable-density, 4½-layer model
Schematic diagram of the structure of a 4½-layer model used to study biophysical interactions in the Arabian Sea.
mixed layer
diurnal thermocline
seasonal thermocline
1) The “diurnal thermocline” layer is included to remember the properties of the water that are left behind when the mixed layer thins during the day.
3) Note that layers all have minimum allowable thicknesses.
main thermocline

32. Variable-density, 6½-layer model 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.
mixed layer
diurnal thermocline
seasonal thermocline
main thermocline upper OMZ
lower OMZ
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.
sub-OMZ layer

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

35. 2-layer model
If the circulation extends to the ocean bottom, a 2-layer model is useful. Its equations are
for layer 1,
for layer 2, and the pressure gradients are

36. 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

37. 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

38. 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.

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

40. 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.
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.

41. 4½-layer model
Meridional section from a solution to a 4½-layer model of the Pacific Ocean, illustrating its layer structure across the central basin.
thermocline
SPLTW
NPIW
AAIW
Note how realistic the density structure is.
In this solution, there is not a realistic, equatorial thermostad. We fixed that deficiency in our TJ study.
Water can transfer between layers with across-interface velocities wi.