1. Wind-forced solutions:
Interior ocean
A short course on:
Modeling IO processes and phenomena
INCOIS
Hyderabad, India
November 16−27, 2015

2. References
(HIG Notes) McCreary, J.P., 1980: Modeling wind-driven ocean circulation. JIMAR , HIG 80-3, Univ. of Hawaii, Honolulu, 64 pp.
InteriorNotes.pdf
Shankar_notes/interior_ocean.pdf

3. Introduction Overview: directly forced & wave responses
Ekman drift & inertial oscillations
Interior-ocean equations
Ekman pumping (β = 0)
Rossby waves and adjustment to Sverdrup balance (β ≠ 0)
Western boundary currents

4. Overview: directly forced & wave responses

5. Mode equations
Equations of motion for a single mode of the LCS model are
The complete 3-d fields are then given by
In this lecture, though, we only discuss single-mode solutions. (In subsequent lectures, we discuss 3-d solutions.)
Solutions can be viewed as having two parts: 1) directly forced and 2) wave parts. Mathematically, the two parts correspond to the particular and homogeneous solutions of a differential equation.
The directly forced response is confined to the region of the wind, and so only impacts the ocean “locally.” The wave response, which can radiate far from the forcing region, affects the ocean “remotely.”

6. Forced vn equation
For convenience, we drop all mixing terms (A = υh = 0) and drop subscripts n. Then, solving for a single equation in v gives
(1)
Solutions to (1) are difficult to find analytically because f is a function of y and the equation includes y derivatives (the term vyyt). It is possible, however, to find analytic solutions to approximate versions of (1) that avoid these mathematical complexities.
Problem #3: Solve the equations of motion to obtain (1).
In the next parts of this talk, we find solutions to (1) that illustrate fundamental ocean processes in idealized settings. In addition, we show movies of solutions in more complex situations for which analytic solutions are difficult (impossible) to obtain.

7. Ekman drift and inertial oscillations

8. Ekman drift and inertial oscillations (β = 0)
The most fundamental forced motion in the ocean is Ekman drift. In an inviscid, single-mode (or 1½-layer) model, Ekman drift occurs at an angle of 90° to the right (left) of the wind in the northern (southern) hemisphere.
To illustrate this response as simply as possible, we assume that the ocean is unbounded, f is constant, and the forcing is by a spatially uniform τx. Then, the equation (1) simplifies to
(2)
Why is it “okay” to consider spatially uniform winds? Because the typical scale of wind the wind forcing (~500−1000 km) is much greater than the Rossby radius of deformation (R ~ 25−50 km).

9. Ekman drift and inertial oscillations (β = 0)
Suppose the wind switches on at t = 0. We split the solution into a time-independent, particular solution
and a homogeneous solution that satisfies (2) with F = 0
The total solution is then
and A and B are determined by applying initial conditions.

10. Ekman drift and inertial oscillations (β = 0)
Assume that the ocean is at rest before the wind switches, so that appropriate initial conditions are u = v = 0 at t = 0.
We use the v momentum equation to write the boundary condition for u in terms of v. We have
Applying the initial conditions gives
so that

11. Ekman drift and inertial oscillations (β = 0)
The steady-state solution is Ekman drift, but GWs at σ = f are also generated to satisfy the initial conditions.
σ/f
k/α - 1 −1
R/Re
Because β = 0 and there are no coasts, only GWs are possible. Because the wind is spatially uniform, only GWs with k = ℓ = 0 can be excited. According to the disp. rel., the waves with zero wavenumber are inertial waves with σ = f.
If the wind is not spatially uniform, GWs with k > 0 and σ > f can are also excited.

12. Ekman drift and inertial oscillations (β = 0)
To summarize, the solutions for u and v when f is constant are
a steady, southward, Ekman drift plus an inertial oscillation in which the velocity vector rotates clockwise at a single frequency f.

13. Ekman drift and inertial oscillations (β ≠ 0)
To summarize, the solutions for u and v when f is constant are
a steady, southward, Ekman drift plus an inertial oscillation in which the velocity vector rotates clockwise at a single frequency f.
Q: How does this simple response change when β ≠ 0?
A: Frequency f and hence the clockwise rotation of the velocity vector differ at each latitude. Very quickly, convergences (divergences) develop between different latitudes, requiring water to downwell (upwell). This process excites gravity waves with ℓ ≠ 0, and is known as β-dispersion.
Movies B

14. Interior-ocean equations

15. Interior-ocean equations
Equations of motion for the un, vn, and pn fields of a single baroclinic mode are
This linear set of equations, however, is difficult to solve.
As discussed next, this approximation is useful is because it filters out the gravity-wave response. Thus, it only describes the slowly varying parts of the response, that is, the directly forced and Rossby wave (if β ≠ 0) parts of the response.
A useful (and still popular) simplification is to drop the acceleration and damping terms from the momentum equations. In addition, the horizontal viscosity terms are assumed small dropped in the interior ocean, and are only retained to represent western boundary currents.

16. Conditions of validity
Under what conditions is the approximate equation set valid?
For convenience, drop subscripts n, mixing and damping terms, and τy forcing. Then, the v equation for the complete set of equations is
but the v equation for the approximate set lacks the first three terms.
So, the approximation is valid provided the first three terms are small compared to the fourth. That will be true provided that
Note that the inequalities will be valid for atmospheric forcing that is large-scale and slowly varying.
and similarly that
Ly2

17. Distortion of free waves
How does the approximate equation set distort the waves in the model?
To consider only free waves, we neglect forcing in the approximate v equation to get
Assuming a sinusoidal wave form
then gives the linear dispersion relation
So, Rossby waves are non-dispersive in the approximate model, a property that makes solutions easy to find.

18. Distortion of free waves
How does the interior-ocean approx. distort the dispersion curves?
σ/f
k/α - 1 −1
σ/f
k/α - 1 −1
R/Re
It eliminates gravity waves and the Rossby curve becomes a straight line (non-dispersive). Moreover, σ is independent of ℓ, so that the Rossby wave curve is not a bowl in k-ℓ space, but a plane.
When are the waves accurately simulated in the interior model? When kR = k/α << 1, ℓR << 1, and σ/f << 1.
R/2Re

19. Ekman pumping (β = 0)

20. Ekman pumping
Written in terms of a 1½-layer model, the interior-ocean equations are
where
and the damping corresponds to entrainment into or detrainment from the layer.
Solving for a single equation in h gives
where wek is the Ekman-pumping velocity, the rate at which wind curl raises or lowers subsurface isopycnals.

21. Ekman pumping with κ = 0 forced by τx
Suppose the ocean is forced by a zonal wind (i.e., τy = 0) and there is no damping (κ = 0). Then, the solution is
so that h grows continuously in time.
With h known, the zonal and meridional velocities are
a superposition of Ekman and geostrophic currents.
Note that, although the geostrophic flow grows linearly in time, the Ekman flow switches on instantly at t = 0 and thereafter remains constant. It switches on instantly because the interior-ocean equations filter out gravity (inertial) waves.

22. Ekman pumping with κ = 0 forced by τx
How long does it take for the layer bottom to upwell to the surface?
which for the above wind, H = 100 m, and f = 10−4 s−1 is t = 368 days.
For this wind, τxy < 0 north of 2000 km and h thins, and the opposite change happens south of 2000 km. The constant Ekman drift shifts water continually from the northern to the southern half of the domain. Counter-rotating geostrophic gyres spin-up in response to h.

23. Ekman pumping with κ ≠ 0 forced by τx
Suppose the ocean is forced by a zonal wind (i.e., τy = 0) and there is damping (κ ≠ 0). Then, the solution is
so that h stops growing.
Movies C1 and C4.
In steady state, Ekman drift still flows from the northern to the southern half of the domain. Water entrains into the layer in the north to provide a source for the Ekman drift and detrains from the layer in the south to provide a sink.

24. Rossby waves and adjustment to Sverdrup balance (β ≠ 0)

25. Adjustment when β ≠ 0 and κ = 0
Suppose the model ocean allows f to vary (β ≠ 0) and there is no damping (κ = 0). Then, h satisfies
We obtain the solution by splitting it into steady-state (particular, forced) and transient (homogenous, wave) parts
where Λ(x,y) is an as yet unspecified function.
To satisfy the initial condition that h = H at t = 0, we must choose Λ(x,y) = −χ(x,y), so that

26. Initial adjustment
To determine the response a short time after the wind switches on, we expand the Rossby-wave term in a Taylor series about t = 0 to get
Thus, at small times, the response is just Ekman pumping!
The response does not change from Ekman pumping until the Rossby waves have time enough to propagate significantly westward.
t +

27. Final adjustment At longer times the solution for all the fields is
where
A packet of Rossby waves propagates westward.
After their passage, the solution adjusts to a steady-state Sverdrup balance.

28. Solution forced by switched-on τx
Consider the response to a switched-on patch of zonal wind.
β-plane –
f-plane
The initial response is the same as on the f-plane. Ekman flow switches on instantly because gravity waves are filtered out of the system, and wind curl drives Ekman pumping.
This process also takes place in the northern Indian Ocean, for example, accounting for some of the annual cycle of the EICC.
It also takes place in the southern IO at interannual time scales, in response to ENSO- and IOD-related wind anomalies.
Ekman pumping can be so large that the interface below rises to the surface, or to a specificed minimum value. In that case, there is upwelling from the layer below (light green area).
At a given longitude, Ekman pumping continues until the passage of Rossby waves. Because they propagate slowly, the Ekman pumping can be large enough for the bottom of the layer to rise to the surface (light green area). In that case, the solution breaks down, and there must be upwelling from deep ocean.
Subsequently, the westward radiation of Rossby waves adjusts the circulation to Sverdrup balance. For zonal winds, the response is similar to the Ekman pumping response except that h, u, and v are affected across the basin.

29. Solution forced by switched-on τy
Consider the response to a switched-on patch of meridional wind. –
τy forcing
τx forcing
The initial response is the same as on the f-plane. Ekman flow switches on instantly because gravity waves are filtered out of the system, and wind curl drives Ekman pumping.

30. Solution forced by switched-on τy
In contrast, the steady-state, Sverdrup response is very different for meridional winds.
τx forcing
τy forcing
Movies C2, C3, and C5.
The Sverdrup response to forcing by τy does not extend westward, but rather is confined to the region of the wind. Mathematically, it does so because the structure of χ changes
and no longer involves a zonal integral of the wind.

31. Western boundary currents

32. Western-boundary currents
When long-wavelength Rossby waves (LWRWs) propagate to a western-ocean boundary, zonal flow associated with them is channeled into a western-boundary current (WBC).
Without momentum mixing, the LWRWs reflect as a packet of short-wavelength Rossby waves (SWRWs) that continuously thins.
Movie MassSource(300days).fli

33. Western-boundary currents
When long-wavelength Rossby waves (LWRWs) propagate to a western-ocean boundary, zonal flow associated with them is channeled into a western-boundary current (WBC).
Without momentum mixing, the LWRWs reflect as a packet of short-wavelength Rossby waves (SWRWs) that continuously thins.
Movie MassSource(300days).fli
With momentum mixing, the WBC thinning stops (or never appears at all) and its offshore structure adjusts to steady-state profile.
Movies D

34. Western-boundary currents
To find the structure of the western-boundary current, neglect time-dependent and vertical-mixing terms and forcing terms in the equations of motion, and for convenience drop subscripts n.
Solving for a single equation in v then gives

35. Western-boundary currents
(1)
Adopting that boundary-layer assumption that Ly2 » Lx2, we drop all y-derivative terms from the right-hand side of (1).
With only Laplacian mixing (ν = 0), then the solution to (1) is
a Munk layer. This layer oscillates, as well as decays, offshore.
With only Raleigh damping (νh = 0), then the solution to (1) is
a Stommel layer.

36. Western-boundary currents
νh = 5x106 cm2/s
νh = 5x107 cm2/s
νh = 5x105 cm2/s
In Solutions D1 − D3, the WBCs are Munk layers that decay and oscillate offshore.