1. Internal boundary layer scaling

Stommel and Webster (1962) discovered a similarity solution of the thermocline equations with an internal boundary layer that could be interpreted as a model of the subtropical main thermocline. The internal boundary layer marks the base of the wind-driven motion, as the deeper circulation is driven by vertical diffusion of heat through the internal boundary layer. The characteristic thickness of the Stommel–Webster internal boundary layer is κ^1/2_υ, where κ_υ is a constant vertical diffusivity. Originally obtained by a linearized analysis, this scaling was confirmed by Young and Ierley (1986) using matched asymptotic expansions. It contrasts with the κ^1/3_υ thickness dependence that follows from the traditional advective–diffusive scaling of the thermocline equations (Welander 1971).

The κ^1/2_υ boundary layer is evidently not peculiar to the particular similarity form studied by Stommel and Webster (1962) and Young and Ierley (1986). Salmon (1990) showed that a κ^1/2_υ internal boundary layer should generally arise in subtropical-gyre solutions of the thermocline equations, although this result depended on a Taylor-series argument that itself relied on an assumption that the vertical convergence in the boundary layer was constant and nonzero. A related discussion was given by Pedlosky (1979, p. 422). (As a reviewer has pointed out, these same arguments establish that a κ^1/2_υ scaling will generally arise in a stationary boundary layer when the advected scalar is passive rather than active, since constant vertical convergence is then generic.) Samelson and Vallis (1997) suggested that the κ^1/2_υ scaling will arise whenever isotherm slopes in the thermocline are fixed as κ_υ → 0 and reported evidence for a κ^1/2_υ scaling in numerical solutions of a closed-basin planetary geostrophic circulation model, despite clear differences between the horizontal structure of the numerical solutions and the Stommel–Webster similarity solution.

The present contribution should be read as a footnote to the articles cited above. In essence, it is a modest extension of the argument of Salmon (1990), cast in a different form. The starting point is a two-layer solution of the ideal (κ_υ = 0) equations, in which temperature is discontinuous across the interface between the layers and the detailed structure of the wind forcing is not specified. A general internal boundary layer equation is then derived that must be satisfied asymptotically by any smooth solution of the diffusive (κ_υ > 0) thermocline equations that approaches the two-layer ideal solution as κ_υ → 0. This point of view resembles that of Young and Ierley (1986), who interpret the ideal limit of the Stommel–Webster solution as a weak (discontinuous) solution of the continuously stratified ideal thermocline equations. Here, it is generally assumed that the relevant smooth solutions exist, but one explicit example is given.

2. The two-layer limit

Following Welander (1971), the dimensionless continuously stratified steady diffusive thermocline equations may be written

f⁻¹M_zyM_zzxf⁻¹M_zxM_zzyβf⁻²M_xM_zzzκ_υM_zzzz(2.1)

where M(x, y, z) satisfies

M_zp.(2.2)

The temperature T and velocities (u, υ, w) may be written as derivatives of M,

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expressing the hydrostatic, geostrophic, and Sverdrup vorticity balances, where the additional boundary conditions w, M → 0 as z → −∞ (or w = M = 0 at the bottom z = −H_B) have been enforced. The Sverdrup transport relation takes the form

βf⁻²M_xx, y,w_E(2.4)

where w_E is the Ekman vertical velocity at the base of the surface boundary layer.

For general w_E < 0, the ideal (κ_υ = 0) thermocline equations have a two-layer (or “one-and-a-half-layer”) solution, with an upper, moving layer of thickness h(x, y) and uniform temperature T = T₀ overlying a deep motionless layer of temperature T = 0. The thickness of the moving layer is

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where H₀ is the depth of the upper layer at the eastern boundary. For example, such solutions have been considered by Parsons (1969) and Veronis (1973) and are equivalent to a ventilated thermocline (Luyten et al. 1983) with a single moving layer.

Now, for 0 < κ_υ ≪ 1, suppose that there exists a solution of the diffusive thermocline equations (2.1) that matches the two-layer solution (2.5) except near z = −h(x, y) where a smooth transition across an internal boundary layer of finite thickness replaces the discontinuity. The analysis below shows that any such solution must asymptotically satisfy an internal boundary layer equation following from (2.1).

To derive this general internal boundary layer equation, it is convenient to introduce the stretched boundary layer coordinate ζ, where

ζδ⁻¹zhx, y(2.6)

is the vertical distance from z = −h scaled by the unknown boundary layer thickness δ. The appropriate matching conditions on T outside the internal boundary layer are then T → T₀ as ζ → +∞ and T → 0 as ζ → −∞. In order to write the corresponding boundary conditions for M_ζζ in a form that is independent of δ, it is necessary to rescale M in the boundary layer by the substitution

Mx, y, zδ²T₀Ax, y, ζ(2.7)

where the absence of order-1 and order-δ contributions to M is consistent with the requirement that M vanish for z < −h when κ_υ = 0, and with the matching conditions on T. Then M_zz = T₀ A_ζζ, and the boundary conditions for A are

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The matching conditions on w are w → (z/h + 1)w_E = δ(ζ/h)w_E as ζ → +∞ and w → const as ζ → −∞. The first of these matches the wind-driven vertical velocity above the internal boundary layer, while the second will give the diffusively driven abyssal upwelling velocity beneath the boundary layer, which must vanish along with the abyssal M as κ_υ → 0. Since M_x = δT₀(A_ζh_x + δA_x), to first order in δ these are

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consistent with (2.8) and (2.9), where c is a constant and use has been made of (2.5).

With the substitution (2.7), the resulting equation for A is

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Balancing the diffusive term against the leading-order advective terms in (2.12) leads to δ² ∝ κ_υ. Thus, the internal boundary layer in the two-layer limit should generally have thickness proportional to κ^1/2_υ. Presumably, an appropriate solution of (2.12) exists. In general, it is not immediately clear how to solve (2.12). The leading horizontal advective terms contain first-order horizontal derivatives, and appropriate lateral boundary conditions must be determined. Note that it is the need to match to constant temperatures above and below the boundary layer that determines the rescaling (2.7) and ultimately requires δ ∝ κ^1/2_υ.

Since the form (2.7) fixes the isotherm slopes independently of δ (T_x/T_z = h_x, T_y/T_z = h_y, to leading order), this result is consistent with the scaling argument of Samelson and Vallis (1997), who presented numerical evidence for a κ^1/2_υ thickness in the central subtropical gyre of a planetary geostrophic circulation model and suggested that the κ^1/2_υ thickness should arise whenever isotherm slopes are fixed as κ_υ → 0. Because of the relative horizontal uniformity of the fluid immediately above (“subtropical mode water” analog) and below (abyssal fluid) the internal boundary layer in the numerical solutions of Samelson and Vallis (1997), the two-layer model can reasonably serve as an approximation to the numerical solutions near the internal boundary layer, despite the existence of a strongly stratified portion of the ventilated thermocline near the surface.

3. A one-dimensional equation

One might expect that the substitution (2.7) would lead to an asymptotic boundary layer equation involving only ζ derivatives of A in the limit δ → 0, corresponding to the thermodynamic balance wT_z ≈ κ_υT_zz. However, this does not happen. The horizontal advective terms of order δ⁻¹ that arise from the substitution (2.7) vanish identically from (2.12), but the leading-order horizontal advective terms that remain in (2.12) are still of order 1, the same order as the leading-order vertical advective term. Consequently, the thermodynamic balance does not, in general, reduce to wT_z ≈ κ_υT_zz as κ_υ → 0. This might be anticipated from the observation that the vertical velocity itself vanishes in this limit.

Special solutions of (2.12) may still be sought in which horizontal advection does vanish, either identically or to leading order. The matching conditions for A are themselves independent of x and y. In the case of (2.10), this reduction is possible because the Sverdrup relation enforces a proportionality between h_x and w_E/h at each point. This suggests the substitution A(x, y, ζ) = B(ζ) in (2.12), or M(x, y, z) = δ²T₀B(ζ). If the resulting equation for B(ζ) were independent of x and y, then a one-dimensional boundary layer theory would exist in the two-layer limit. This substitution gives

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Comparison with (2.1) shows that this is a vertical advective–diffusive balance in which the term M_x is replaced by M_zh_x, a consequence of fixing the isotherm slopes to first order. The equality in (3.1) can in general be satisfied only if the quantity κ_υf²/(βh_xT₀) is constant since the latter is clearly independent of ζ while by assumption B depends only on ζ. From (2.5),

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but inspection shows that h/w_E is not generally constant. Thus, an asymptotic one-dimensional internal boundary layer theory for the two-layer limit of (2.1) does not exist in general. A solution may still exist that approaches the two-layer solution in the limit κ_υ → 0, and, if it exists, it must have δ ∝ κ^1/2_υ, but the corresponding internal boundary layer will in general not be independent of horizontal position; that is, it will remain intrinsically three-dimensional.

If the expression (3.2) were constant, the equation (3.1) would reduce to the equation ultimately solved by Young and Ierley (1986) in their asymptotic analysis of the similarity solution discovered by Stommel and Webster (1962). This can be seen as follows. Set

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and let

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Then (3.1) is

FFF(3.5)

and the boundary conditions (2.10) and (2.11) are

Fζζ,Fζ(3.6)

where prime denotes derivative with respect to the argument. The two conditions (3.6) are sufficient for the third-order equation (3.5) since the first also imples F′(ζ → +∞) → 1. The abyssal upwelling velocity is determined by the value c, where F(ζ → −∞) → c, since w(ζ → −∞) → cδw_E/h.

The equation (3.5) with the boundary conditions (3.6) is solved by Young and Ierley (1986) in their analysis of the Stommel–Webster similarity solution (with the sign of ζ reversed). Their solution yields c = 0.875 74. Note that the Stommel–Webster similarity solution is not of “two-layer” type: it retains zonal temperature gradients above and below the internal boundary layer even in the limit κ_υ → 0. Near the internal boundary layer, the two-layer model is a more accurate representation of the numerical solutions of Samelson and Vallis (1997) than is the Stommel–Webster similarity solution because of the large zonal temperature gradients in the latter and the dependence of the thermocline depth in the numerical solutions on horizontal position, which roughly follows the two-layer solution.

As an explicit example of a two-layer solution with this type of internal boundary layer, consider (2.5) with H₀ = 0 and w_E = af²(x − x_E), where a is a constant. In this case, w_E/h is constant, and (3.1) is independent of x and y and has the form (3.5), with boundary conditions (3.6). This is a two-layer solution with a one-dimensional internal boundary layer that has thickness δ ∝ κ^1/2_υ and is independent of horizontal position. In this case, the thermodynamic balance in the boundary layer does reduce to wT_z ≈ κ_υT_zz.

4. Related examples

Salmon and Hollerbach (1991) obtained some special solutions of the thermocline equations that are relevant to the present discussion. In one class of solutions (their “S₁₂”), the temperature changed abruptly across an internal boundary layer, as in the solution discussed above. In a second class of solutions (their “S₁₃”), the potential vorticity changed abruptly across an internal boundary layer, while the temperature field remained smooth as κ_υ → 0. For each of these classes, they presented specific examples for the special case in which the boundary layer is located at a constant depth, independent of horizontal position [their Eqs. (8.1)–(8.9) and (8.10)–(8.12), respectively]. In both of these specific examples, the thickness of the internal boundary layer was κ^1/2_υ, as in the solution discussed above.

It is especially interesting that a κ^1/2_υ boundary layer appears also in their second example, for which the jump is in potential vorticity rather than temperature. An extrapolation of the argument given above would seem to suggest that the boundary layer thickness should be proportional to κ^1/3_υ, as in the traditional advective–diffusive scaling, since the potential vorticity fT_z = fM_zzz (rather than temperature T = M_zz) must be matched outside the boundary layer, and this would appear to lead to a factor δ³ in the scaling (2.7). In this case, however, M in the boundary layer may have contributions of zero, first, and second order in δ, along with the third-order term associated with the potential vorticity matching. Thus, the simple extrapolation is misleading, and the boundary layer again scales with κ^1/2_υ.