## 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}*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}

The *κ*^{1/2}_{υ}*κ*^{1/2}_{υ}*κ*^{1/2}_{υ}*κ*^{1/2}_{υ}*κ*_{υ} → 0 and reported evidence for a *κ*^{1/2}_{υ}

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

*f*

^{−1}

*M*

_{zy}

*M*

_{zzx}

*f*

^{−1}

*M*

_{zx}

*M*

_{zzy}

*β*

*f*

^{−2}

*M*

_{x}

*M*

_{zzz}

*κ*

_{υ}

*M*

_{zzzz}

*M*(

*x, y, z*) satisfies

*M*

_{z}

*p.*

*T*and velocities (

*u, υ, w*) may be written as derivatives of

*M,*

*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*

^{−2}

*M*

_{x}

*x, y,*

*w*

_{E}

*w*

_{E}is the Ekman vertical velocity at the base of the surface boundary layer.

*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*=

*T*

_{0}overlying a deep motionless layer of temperature

*T*= 0. The thickness of the moving layer is

*H*

_{0}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).

*ζ,*where

*ζ*

*δ*

^{−1}

*z*

*h*

*x, y*

*z*= −

*h*scaled by the unknown boundary layer thickness

*δ.*The appropriate matching conditions on

*T*outside the internal boundary layer are then

*T*→

*T*

_{0}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

*M*

*x, y, z*

*δ*

^{2}

*T*

_{0}

*A*

*x, y, ζ*

*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}=

*T*

_{0}

*A*

_{ζζ}, and the boundary conditions for

*A*are

*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}=

*δT*

_{0}(

*A*

_{ζ}

*h*

_{x}+

*δA*

_{x}), to first order in

*δ*these are

*c*is a constant and use has been made of (2.5).

*A*is

*δ*

^{2}∝

*κ*

_{υ}. Thus, the internal boundary layer in the two-layer limit should generally have thickness proportional to

*κ*

^{1/2}

_{υ}

*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}*κ*^{1/2}_{υ}*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*_{υ}*T*_{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*_{υ}*T*_{zz} as *κ*_{υ} → 0. This might be anticipated from the observation that the vertical velocity itself vanishes in this limit.

*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*) =

*δ*

^{2}

*T*

_{0}

*B*(

*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

*M*

_{x}is replaced by

*M*

_{z}

*h*

_{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*

_{υ}

*f*

^{2}/(

*βh*

_{x}

*T*

_{0}) is constant since the latter is clearly independent of

*ζ*while by assumption

*B*depends only on

*ζ.*From (2.5),

*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}

*FF*

*F*

*F*

*ζ*

*ζ,*

*F*

*ζ*

*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 *H*_{0} = 0 and *w*_{E} = *af*^{2}(*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}*wT*_{z} ≈ *κ*_{υ}*T*_{zz}.

*δ*from (3.3) is constant, as in the preceding example, it may still provide a useful approximation to

*M*if

*δ*is only approximately constant. If

*δ*

_{x}

*δ*

_{y}

*δ*

*δ,*

**∇***w*

_{E}

*h*

*w*

_{E}

*h*

*δ.*

*w*

_{z}in the two-layer solution is horizontally uniform. This requires that the fractional variations in

*w*

_{E}and

*h*either be separately small or cancel to first order. The condition (3.7) can be recast in terms of

*w*

_{E}and

*H*

_{0}using (2.5). If

*w*

_{E}is independent of

*x*and

*y,*and

*H*

_{0}is sufficiently large, then the approximation will be accurate. If

*w*

_{E}is independent of

*x,*and

*h*(

*x*) −

*H*

_{0}≈

*H*

_{0}, then the approximation may be inaccurate.

## 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 “*S*_{12}”), the temperature changed abruptly across an internal boundary layer, as in the solution discussed above. In a second class of solutions (their “*S*_{13}”), 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}

It is especially interesting that a *κ*^{1/2}_{υ}*κ*^{1/3}_{υ}*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 *δ*^{3} 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}

## Acknowledgments

This research was supported by the National Science Foundation, Division of Ocean Sciences (Grants OCE94-15512 and OCE98-96184). I am grateful for comments from J. Pedlosky and R. Salmon.

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