The Ventilated Thermocline

The ’LPS’ model of the ventilated thermocline, developed in Luyten et al. (1983)1, gives another explanation for how lower layers in a stratified ocean can be in motion even if not exposed directly to wind forcing (that is, in addition to the explanation given by Rhines & Young which we’ve looked at earlier). In doing so the model also predicts qualitative features of the upper-ocean density stratification. Using the layered plane- tary geostrophic (PG) equations rather than the quasi-geostrophic (QG) equations, the model solves for mean layer thicknesses, i.e. the stratification, rather than having them specified as done in QG. The key ingredient of the model is a north-south density contrast where some dense layers outcrop at the surface—further poleward than light layers. Figure 1 shows a simplest-possible model that consist of three layers: a light layer 1 which resides at the surface at low latitudes, then a somewhat denser layer 2 which outcrops at latitude y2 and, finally, an even denser deep layer 3 which doesn’t outcrop anywhere and is at rest. We may assume that the deepest layer is at rest either because it’s not being forced anywhere or, perhaps more realistically, because it is very deep (so that velocites are very small even if they are not identically zero). The model assumes that the density of the layers (ρ1, ρ2 and ρ3) and the outcrop latitude y2 are specified, and a surface

Ekman pumping velocity wE (made here to only depend on latitude, for simplicity) is also given. Geostrophic velocities in the layers are found from thermal wind:

g u = kˆ × ∇η, (1) 1 f g g0 u = kˆ × ∇η − 12 kˆ × ∇h , (2) 2 f f 1 g g0 g0 u = kˆ × ∇η − 12 kˆ × ∇h − 23 kˆ × ∇h. (3) 3 f f 1 f

Here η is the sea surface height, h1 and h2 are the thicknesses of layers 1 and 2 so that

1Luyten, R. J., J. Pedlosky & H. Stommel (1983): The ventilated thermocline, Journal of Physical Oceanography, 13, 292–309.

1 Figure 1: The 2-layer (+ one deep layer at rest) model for the ventilated thermocline. From Vallis’ book (Fig. 20.17).

h = h1 + h2 is the total depth reached by the wind-driven ﬂow (remember, the lower 0 0 layer is at rest) and, ﬁnally, g12 and g23 are the reduced gravities of the two interfaces.

Note that the expressions above for u2 and u3 have assumed a rigid upper lid where we ignore the sea surface height contribution when calculating layer thicknesses (but we still use it to calculate the barotropic pressure gradient). Since we assume that the lower layer is at rest, i.e. u3 = 0, the relationships for the ﬂow in the two layers above become (integrating up from the bottom):

g0 u = 23 kˆ × ∇h, (4) 2 f g0 g0 u = 23 kˆ × ∇h + 12 kˆ × ∇h (5) 1 f f 1 g0 = u + 12 kˆ × ∇h . (6) 2 f 1

2 Sverdrup transport north of y2 The ﬂow regimes in this model naturally separate into two main regions: one north and another south of latitude y2 where layer 2 outcrops (we assume we are on the northern hemisphere). North of y2 all the wind-driven transport is carried by layer 2 (layer 1 is absent here and layer 3 is at rest). In terms of the depth-integrated steady and linear vorticity equation, this can be written:

βh2v2 = f (wE − wb) , (7) where we have used the continuity equation and then assumed that the vertical velocity at the top of the layer is the vertical velocity out of the surface Ekman layer. The vertical velocity at the bottom is given by the kinematic boundary condition, and since the lower layer is at rest this becomes

wb = u2 · ∇h g0 ! = 23 kˆ × ∇h · ∇h f = 0. (8)

This last result also applies south of y2 and, in fact, it is a quite general relationship: When the wind-driven geostrophic flow is confined to the upper ocean, sheltered from any non-flat bottom topography by deeper layers at rest, then the depth-integrated meridional transport V is given by

βV = fwE, (9) an expression we have used multiple times earlier in this course. So we have

βh2v2 = fwE, (10)

3 or, if inserting our expression for the geostrophic velocity in layer 2 and noting that north of y2 we have h2 = h, g0 ∂h βh 23 = fw . (11) f ∂x E This can be re-written as ∂ h2 ! f 2 = 0 wE, (12) ∂x 2 βg23 and if we now integrate this expression along a latitude line, from location x to the eastern boundary at xe, we arrive at

2 2 2 h (x) = He + D0(x), (13) where 2 xe 2 2f D0(x) = − 0 wE dx (14) βg23 ˆx and where He = h(xe) is the thickness at the eastern boundary. Note that He must be a constant to avoid ﬂow through the eastern boundary (we assume the boundary runs north-south). It will have to remain a free parameter in our problem, to be speciﬁed by other independent dynamical considerations or from observations. But given He

(and, of course, the Ekman pumping velocity wE), we have a complete solution north of outcrop latitude y2.

From this last expression we can see that for Ekman suction, wE > 0, typical for a subpolar gyre, the layer thickness decreases westward and the solution is only valid 2 2 up until position x where h(x) = 0 (so where Do = −He ). West of this position layer 2 would also vanish and layer 3 would be exposed to the winds—in violation of the assumptions we have made for this model. But for conditions over a subtropical gyre, where wE < 0, we get a layer thickness that increases progressively away from the eastern boundary. We then have v2 ∝ ∂h/∂x < 0, i.e. a southward ﬂow. So water columns in layer 2 will move southward, and upon reaching latitude y2 they will subduct, meaning they will slide underneath layer 1 (layer 2 is, after all, denser than layer 1). From here on layer-2 columns are no longer forced directly by the winds. Do they stop moving? If there is no friction acting on them (or, as is more realistic, friction is very

4 small in the upper ocean), then why should they?

Sverdrup transport south of y2

Now let’s apply the Sverdrup relation to the region south of y2. Even though it is only layer 1 that is now directly forced by the wind we integrate over both layers to get

β (h2v2 + h1v1) = fwE, (15) or, if we wish,

β [(h − h1) v2 + h1 (v2 + ∆v)] = fwE, (16)

0 where ∆v = (g12/f) ∂h1/∂x is the velocity jump from layer 2 to layer 1. The second and third terms cancel, and if we now insert our expressions for the geostrophic velocities we get g0 ∂h g0 ∂h ! β h 23 + h 12 1 = fw , (17) f ∂x 1 f ∂x E or 2 0 2 ! 2 ∂ h g12 h1 f + 0 = 0 wE. (18) ∂x 2 g23 2 βg23

Integrating from location x to the eastern boundary at xe gives

0 2 g12 2 2 2 2 h (x) + 0 h1(x) = He + H1e + D0(x), (19) g23 where He and D0(x) are the same as before and where H1e = h1(xe) is the thickness of layer 1 on the eastern boundary. This has to be constant for the same reason that He is a constant (no ﬂow through the boundary) and, actually, it must be zero since it was zero at y2 (layer 1 vanishes there). So the integrated Sverdrup relation south of y2 is

0 2 g12 2 2 2 h (x) + 0 h1(x) = He + D0(x). (20) g23

This integrated 2-layer Sverdrup relation holds everywhere south of y2. But there are two unknowns, the total thickness of the active layers (h) and upper-layer thickness

5 (h1). So to make progress we need to bring in additional dynamical considerations, and those considerations are diﬀerent in what turns out to be three dynamically distinct regions south of y2.

The region south of y2 ﬁlled with subducted layer-2 waters

First to the region south of y2 where the subducted layer-2 water columns ﬂow. What additional dynamical considerations can we bring in here? Well, one thing we can say about these subducted columns is that if there are no other processes aﬀecting them (e.g. no friction), then the columns will conserve their PV. In other words,

D f ! = 0, (21) Dt h2 which in steady state means f u2 · ∇ = 0, (22) h2 or g0 ! f 23 kˆ × ∇h · ∇ = 0. (23) f h2 This implies that contours of h—which, remember, are streamlines for the ﬂow in layer

2—coincide with contours of q2. We can therefore write

f q2 = = G2(h), (24) h2 where G2 is some function relating q2 to h. As it turns out, this functional relationship can be found at the outcrop latitude y2 where f = f(y2) = f2 and where h2 = h. So we have, at y2, f f2 q2 = = . (25) h2 h

And the fortunate thing is that this functional relationship also holds southward of y2, at least in regions traversed by water column that subducted at y2. Given this, then, we can deduce the thicknesses of both layers as fractions of the total thickness h of the

6 ﬂow, i.e.

f h2 = h, (26) f2 f ! h1 = h − h2 = 1 − h. (27) f2

To repeat, these relationships are valid in regions south y2 traversed by layer-2 water columns that were previously ventilated north of y2. So now we have an expression for h1 that we can insert into the Sverdrup relation (20) to arrive at

0 " ! #2 2 g12 f 2 2 h (x) + 0 1 − h(x) = He + D0(x), (28) g23 f2 or H2 + D2(x) h2(x) = e 0 . (29) 0 2 g12 f 1 + 0 1 − g23 f2 From this expression then, individual layer thicknesses and geostrophic velocities can be calculated.

A sample solution from an idealized subtropical gyre (wE < 0) is shown in Figure 2. The total layer thickness h (left panel) acts as streamlines for the flow in layer 2 and shows that the flow in this layer is southwards everywhere, both where the layer is forced directly by the winds (north of y2) and where it resides under layer 1 (south of y2). The upper layer thickness h1 (right panel) does not by itself act as streamlines for the flow in that layer, but it is clear that the flow in this layer is also southward. The upper layer, of course, does not exist north of y2. The figure also shows two regions that are grayed out; there is a a ’shadow zone’ in the south-east and a ’pool’ region in the north-west. We will look at each region in turn, but it should be clear from the figure that in neither region do streamlines for layer 2 reach the outcrop latitude y2. This means, of course, that within these two regions we can no longer assume that layer-2 PV is set at y2 (i.e. from functional relationship 25). To find the flow here we’ll need to think differently.

7 Figure 2: A sample solution of the 2-layer problem in a domain of normalized dimensions [0, 1] with the outcrop latitude y2 = 0.8. The left panel shows the total thickness of the active layers (h) while the right panel shows the upper layer thickness (h1). The shaddow zone near the eastern wall and the western pool are regions where layer 2 is ﬁlled with waters that haven’t been exposed to the ventilated region north of y2. From Vallis’ book (Fig. 20.18).

8 The shadow zone

Figure 2 contains two dashed red curves, each marking out the boundary of the western pool and the shadow zone. The dashed line in the east is the easternmost streamline for layer 2 that manages to reach y2 before the eastern wall is met. No streamlines east of this one are able to reach y2 without running some distance along the eastern wall.

But layer-2 ﬂow along the eastern wall is impossible south of y2 for the following reason: The water columns are unforced (they are underneath layer 1) and therefore conserve q2 = f/h2. So a north-south motion along the wall would have to obey v·∂ (f/h2) /∂y =

0. And since f would change for such meridional motion, so would h2. But ∂h2/∂y 6= 0 would lead to ﬂow into or out of the wall, clearly in violation with the kinematic boundary condition there. So we simply cannot have any unforced layer-2 ﬂow along the eastern wall. One could imagine that all streamlines in the shadow zone meet at

(xe, y2), never actually touching the wall (until safely at the outcrop line). But this would set up extremely large geostrophic ﬂows near (xe, y2), a possibility for which there is absolutely no observational evidence. We have to conclude that layer 2 is most likely stagnant in the shadow zone. If this is indeed the case, i.e. if u2 = 0 in the shadow zone, then all the wind-driven transport is carried in layer 1. We then have

βh1v1 = fwE, (30) or g0 ∂h βh 12 1 = fw . (31) 1 f ∂x E The solution, after integrating from a location x (within the shadow zone) to the eastern wall, is

2 xe 2 2f h1(x) = − 0 wEdx βg12 ˆx 0 g23 2 = 0 D0(x), (32) g12

9 where we have again used the fact that H1e = h1(xe) = 0 (since it is zero at y2).

The actual boundary of the shadow zone, described by (xs, y) where xs = xs(y), can be determined from the observation that this boundary is a streamline and hence a constant-h curve. And that streamline ends up at (xe, y2) where h = He. So since this streamline is the last one along which layer-2 columns that ventilate at y2 traverse, we can use the expression valid for ventilated layer-2 columns and write

H2 + D2(x ) e 0 s = H2, (33) 0 2 e g12 f 1 + 0 1 − g23 f2 or, 0 !2 2 g12 f 2 D0(xs) = 0 1 − He . (34) g23 f2 So we have 2 xe 0 !2 2f g12 f 2 − 0 wEdx = 0 1 − He , (35) βg23 ˆxs g23 f2 and if the Ekman pumping is only varying with latitude (a convenient simpliﬁcation, of course), then 2 0 !2 2f g12 f 2 − 0 wE (xe − xs) = 0 1 − He , (36) βg23 g23 f2 which gives the ﬁnal result

0 2 g12 f 2 0 1 − He g23 f2 xs(y) = xe + 2f 2 . (37) 0 wE βg23

The western pool

As Figure 2 suggests, there may also be a ’pool’ region in the north-western part of the domain where layer-2 streamlines do not reach the outcrop latitude y2. So water columns in this regions are unventilated. Are they also therefore stagnant, as in the shadow zone? It could be. But an important diﬀerence from the shadow zone is that the pool region is connected to a western boundary layer (which we don’t tackle in this discussion). In contrast to the situation at the eastern wall, layer-2 water columns can travel along the

10 western wall without being forced by the wind since friction is able to balance the change in PV from the latitudional variation they experience. So the arguments for a stagnant layer 2 in the western pool are not as compelling. Instead, Luyten et al. suggests that layer 2 is ’forced’ by baroclinic instability and eddy motion that homogenizes PV, as in Rhines & Young theory. Let’s examine what such a possibility leads to. Assuming, then, that the western pool experiences baroclinic instability that manages to homogenize layer-2 PV, we set

q2,p = const, (38)

and then take that constant from the q2 value at the boundary of the pool. More speciﬁcally, we take it from the value at the point where the boundary runs into the western boundary at latitude y2, i.e. at point (xw, y2). So we set

f(y2) q2,p = , (39) h2(xw, y2) where the layer thickness at (xw, y2) is found from the integrated Sverdrup relation along y2 using the 1-layer expression (the one valid north of and down to y2):

2 2 2 2 h2(xw, y2) = h (xw, y2) = He + D0(xw, y2), (40) where 2 xe 2 2f2 D0(xw, y2) = − 0 wE(y2) dx. (41) βg23 ˆxw

Then, having determined q2,p, we now have (everywhere in the pool region):

f h2 = (42) q2,p f h1 = h − h2 = h − . (43) q2,p

Note that in the pool region h2 only varies in the meridional direction whereas h1 can also vary zonally (since h can vary zonally). Equipped with these relations we can now

11 ﬁnd the solution in the pool region from the zonally-integrated Sverdrup relation for regions south of y2, i.e. from (20). Substituting in the expression for h1, gives

0 !2 2 g12 f 2 2 h + 0 h − = He + D0(x, y), (44) g23 q2,p or 0 ! 0 0 !2 g12 2 f g12 g12 f 2 2 1 + 0 h − 2 0 h + 0 − He − D0(x) = 0. (45) g23 q2,p g23 g23 q2,p 0 0 Introducing Γ = g12/g23, the solution to this second-order equation becomes

21/2 f (1 − Γ) [H2 + D2(x)] − Γ f Γ e 0 q2,p h(x) = q2,p + . (46) (1 + Γ) (1 + Γ)

The positive root is chosen as this is the one that will lead to a continuous deepening of the total wind-driven flow westward (as we expect for wE < 0). Finally, we can also find an expression for the boundary of the western pool by noting that h on this boundary takes on the same value as at (xw, y2). If we approach this boundary from the east, i.e. from the region filled with ventilated layer-2 waters, we can then write H2 + D2(x ) e 0 p = H2 + D2(x , y ), (47) 0 2 e 0 w 2 g12 f 1 + 0 1 − g23 f2 where xp = xp(y) is the zonal position of the boundary as a function of latitude. Again 0 0 resorting to the shorthand Γ = g12/g23, we thus find xp from

!2  !2 2 f f 2 D0(xp) = Γ 1 − + 1 + Γ 1 −  D0(xw, y2), (48) f2 f2 or   2 xe !2 !2 2f f f 2 − 0 wEdx = Γ 1 − + 1 + Γ 1 −  D0(xw, y2). (49) βg23 ˆxs f2 f2

12 For the idealized case of winds that only vary meridionally, we get an easy expression:

2 2 f f 2 Γ 1 − + 1 + Γ 1 − D (xw, y2) f2 f2 0 xs(y) = xe + 2f 2 . (50) 0 wE βg23

Summing it all up

The ’LPS’ model of Luyten et al. is of course an extremely simpliﬁed version of the real thing, designed so that analytic solutions could be found. But it’s important to pull out the essensial qualitative features that it predicts. Here’s what I personally take home:

1. Deeper density layers may be in motion in a wind-driven subtropial gyre if those layers were once exposed to the atmosphere—ventilated—somewhere else. The LPS model finds the flow in such layers by assuming that water columns conserve the PV that they have at the location where they subduct below lighter waters. That deeper layers in e.g. the North Atlantic and North Pacific have indeed been ventilated can be seen from observations of the age distribution of tracers like oxygen mapped onto various density surfaces.

2. The theory predicts the existence of a ’shadow zone’ that extends westward from the eastern boundary where the waters in deeper layers have not been ventilated further north. The LPS model assumes that the deeper layers here are stagnant, and the zone is thus dynamically similar to the eastern shadow zone in Rhines & Young theory in that motion is likely shut down there by baroclinic Rossby waves transmitting information about the no-normal ﬂow boundary condition of the eastern boundary.

3. The theory also shares another similarity with Rhines & Young theory: a ’western pool’ region in the north-west part of the domain, where the model needs to be joined with a western boundary current. The Ekman pumping deforms the PV contours of the wind-driven ﬂow (total thickness h) to the extent that some contours here wrap back into the western boundary layer before reaching the outcrop latitude for layer-2 waters. So the water in layer 2 here is neither driven

13 directly by winds nor retaining motion set up by outcrop further north. Instead, as with Rhines & Young theory, Luyten et al. assumes that this is a region where lower layers are energized by baroclinic instability and eddy homogenization of lower-layer PV.

Looking for observational or numerical model evidence in support (or contradiction!) of Luyten et al’s model will be left for the second home work assignment.