The diffusivity dependence of internal boundary layers in solutions of the continuously stratified, diffusive
thermocline equations is revisited. If a solution exists that approaches a two-layer solution of the ideal thermocline
equations in the limit of small vertical diffusivity kᵥ, it must contain an internal boundary layer that collapses
to a discontinuity as kᵥ → 0. An asymptotic internal boundary layer equation is derived for this case, and the
associated boundary layer thickness is proportional to kᵥ¹/². In general, the boundary layer remains three-dimensional and the thermodynamic equation does not reduce to a vertical advective–diffusive balance even as the
boundary layer thickness becomes arbitrarily small. If the vertical convergence varies sufficiently slowly with
horizontal position, a one-dimensional boundary layer equation does arise, and an explicit example is given for
this case. The same one-dimensional equation arose previously in a related analysis of a similarity solution that
does not itself approach a two-layer solution in the limit kᵥ → 0

Samelson, R. M.

Undetermined

ScholarsArchive@OSU

AUGUST 1999 2099N O T E S A N D C O R R E S P O N D E N C Eq 1999 American Meteorological SocietyNOTES AND CORRESPONDENCEInternal Boundary Layer Scaling in ‘‘Two Layer’’ Solutions of theThermocline EquationsR. M. SAMELSONCollege of Oceanic and Atmospheric Sciences, Oregon State University, Corvallis, Oregon12 May 1998 and 21 December 1998ABSTRACTThe diffusivity dependence of internal boundary layers in solutions of the continuously stratified, diffusivethermocline equations is revisited. If a solution exists that approaches a two-layer solution of the ideal thermoclineequations in the limit of small vertical diffusivity ky , it must contain an internal boundary layer that collapsesto a discontinuity as ky → 0. An asymptotic internal boundary layer equation is derived for this case, and theassociated boundary layer thickness is proportional to . In general, the boundary layer remains three-dimen-1/2kysional and the thermodynamic equation does not reduce to a vertical advective–diffusive balance even as theboundary layer thickness becomes arbitrarily small. If the vertical convergence varies sufficiently slowly withhorizontal position, a one-dimensional boundary layer equation does arise, and an explicit example is given forthis case. The same one-dimensional equation arose previously in a related analysis of a similarity solution thatdoes not itself approach a two-layer solution in the limit ky → 0.1. Internal boundary layer scalingStommel and Webster (1962) discovered a similaritysolution of the thermocline equations with an internalboundary layer that could be interpreted as a model ofthe subtropical main thermocline. The internal boundarylayer marks the base of the wind-driven motion, as thedeeper circulation is driven by vertical diffusion of heatthrough the internal boundary layer. The characteristicthickness of the Stommel–Webster internal boundarylayer is , where ky is a constant vertical diffusivity.1/2kyOriginally obtained by a linearized analysis, this scalingwas confirmed by Young and Ierley (1986) usingmatched asymptotic expansions. It contrasts with thethickness dependence that follows from the tradi-1/3kytional advective–diffusive scaling of the thermoclineequations (Welander 1971).The boundary layer is evidently not peculiar to1/2kythe particular similarity form studied by Stommel andWebster (1962) and Young and Ierley (1986). Salmon(1990) showed that a internal boundary layer should1/2kygenerally arise in subtropical-gyre solutions of the ther-mocline equations, although this result depended on aTaylor-series argument that itself relied on an assump-Corresponding author address: Dr. Roger M. Samelson, Collegeof Oceanic and Atmospheric Sciences, 104 Ocean AdministrationBuilding, Oregon State University, Corvallis, OR 97331-5503.E-mail: rsamelson@oce.orst.edution that the vertical convergence in the boundary layerwas constant and nonzero. A related discussion was giv-en by Pedlosky (1979, p. 422). (As a reviewer has point-ed out, these same arguments establish that a scaling1/2kywill generally arise in a stationary boundary layer whenthe advected scalar is passive rather than active, sinceconstant vertical convergence is then generic.) Samelsonand Vallis (1997) suggested that the scaling will1/2kyarise whenever isotherm slopes in the thermocline arefixed as ky → 0 and reported evidence for a scaling1/2kyin numerical solutions of a closed-basin planetary geo-strophic circulation model, despite clear differences be-tween the horizontal structure of the numerical solutionsand the Stommel–Webster similarity solution.The present contribution should be read as a footnoteto the articles cited above. In essence, it is a modestextension of the argument of Salmon (1990), cast in adifferent form. The starting point is a two-layer solutionof the ideal (ky 5 0) equations, in which temperatureis discontinuous across the interface between the layersand the detailed structure of the wind forcing is notspecified. A general internal boundary layer equation isthen derived that must be satisfied asymptotically byany smooth solution of the diffusive (ky . 0) ther-mocline equations that approaches the two-layer idealsolution as ky → 0. This point of view resembles thatof Young and Ierley (1986), who interpret the ideal limitof the Stommel–Webster solution as a weak (discontin-uous) solution of the continuously stratified ideal ther-mocline equations. Here, it is generally assumed that2100 VOLUME 29J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Ythe relevant smooth solutions exist, but one explicit ex-ample is given.2. The two-layer limitFollowing Welander (1971), the dimensionless con-tinuously stratified steady diffusive thermocline equa-tions may be written2 f 21Mzy Mzzx 1 f 21Mzx Mzzy 1 b f 22Mx Mzzz 5 ky Mzzzz,(2.1)where M(x, y, z) satisfiesMz 5 p. (2.2)The temperature T and velocities (u, y , w) may be writ-ten as derivatives of M,21 21T 5 M , u 5 2 f M , y 5 f M ,zz zy zx22w 5 b f M , (2.3)xexpressing the hydrostatic, geostrophic, and Sverdrupvorticity balances, where the additional boundary con-ditions w, M → 0 as z → 2` (or w 5 M 5 0 at thebottom z 5 2HB) have been enforced. The Sverdruptransport relation takes the formbf 22Mx(x, y, 0) 5 wE, (2.4)where wE is the Ekman vertical velocity at the base ofthe surface boundary layer.For general wE , 0, the ideal (ky 5 0) thermoclineequations have a two-layer (or ‘‘one-and-a-half-layer’’)solution, with an upper, moving layer of thicknessh(x, y) and uniform temperature T 5 T0 overlying a deepmotionless layer of temperature T 5 0. The thicknessof the moving layer is1/2x22 f2h(x, y) 5 H 1 w (x, y) dx , (2.5)0 E E1 2bT0 xewhere H0 is the depth of the upper layer at the easternboundary. For example, such solutions have been con-sidered by Parsons (1969) and Veronis (1973) and areequivalent to a ventilated thermocline (Luyten et al.1983) with a single moving layer.Now, for 0 , ky K 1, suppose that there exists asolution of the diffusive thermocline equations (2.1) thatmatches the two-layer solution (2.5) except near z 52h(x, y) where a smooth transition across an internalboundary layer of finite thickness replaces the discon-tinuity. The analysis below shows that any such solutionmust asymptotically satisfy an internal boundary layerequation following from (2.1).To derive this general internal boundary layer equa-tion, it is convenient to introduce the stretched boundarylayer coordinate z, wherez 5 d21(z 1 h(x, y)) (2.6)is the vertical distance from z 5 2h scaled by the un-known boundary layer thickness d. The appropriatematching conditions on T outside the internal boundarylayer are then T → T0 as z → 1` and T → 0 as z →2`. In order to write the corresponding boundary con-ditions for Mzz in a form that is independent of d, it isnecessary to rescale M in the boundary layer by thesubstitutionM(x, y, z) 5 d2T0A(x, y, z), (2.7)where the absence of order-1 and order-d contributionsto M is consistent with the requirement that M vanishfor z , 2h when ky 5 0, and with the matching con-ditions on T. Then Mzz 5 T0 Azz, and the boundaryconditions for A are1 as z → 1` (2.8)A →zz 50 as z → 2`. (2.9)The matching conditions on w are w → (z/h 1 1)wE 5d(z/h)wE as z → 1` and w → const as z → 2`. Thefirst of these matches the wind-driven vertical velocityabove the internal boundary layer, while the second willgive the diffusively driven abyssal upwelling velocitybeneath the boundary layer, which must vanish alongwith the abyssal M as ky → 0. Since Mx 5 dT0(Azhx 1dAx), to first order in d these arez as z → 1` (2.10)A →z 5c as z → 2`, (2.11)consistent with (2.8) and (2.9), where c is a constantand use has been made of (2.5).With the substitution (2.7), the resulting equation forA is21f [h (A A 2 A A ) 2 h (A A 2 A A )]x zzy zz zzz zy y zzx zz zzz zx22 21 221 b f h A A 1 d[ f (A A 2 A A ) 1 b f A A ]x z zzz zx zzy zy zzx x zzz22 215 d T k A .0 y zzzz (2.12)Balancing the diffusive term against the leading-orderadvective terms in (2.12) leads to d2 } ky . Thus, theinternal boundary layer in the two-layer limit shouldgenerally have thickness proportional to . Presum-1/2kyably, an appropriate solution of (2.12) exists. In general,it is not immediately clear how to solve (2.12). Theleading horizontal advective terms contain first-orderhorizontal derivatives, and appropriate lateral boundaryconditions must be determined. Note that it is the needto match to constant temperatures above and below theboundary layer that determines the rescaling (2.7) andultimately requires d } .1/2kySince the form (2.7) fixes the isotherm slopes inde-pendently of d (Tx/Tz 5 hx, Ty/Tz 5 hy, to leading order),this result is consistent with the scaling argument ofSamelson and Vallis (1997), who presented numericalevidence for a thickness in the central subtropical1/2kygyre of a planetary geostrophic circulation model andsuggested that the thickness should arise whenever1/2kyAUGUST 1999 2101N O T E S A N D C O R R E S P O N D E N C Eisotherm slopes are fixed as ky → 0. Because of therelative horizontal uniformity of the fluid immediatelyabove (‘‘subtropical mode water’’ analog) and below(abyssal fluid) the internal boundary layer in the nu-merical solutions of Samelson and Vallis (1997), thetwo-layer model can reasonably serve as an approxi-mation to the numerical solutions near the internalboundary layer, despite the existence of a strongly strat-ified portion of the ventilated thermocline near the sur-face.3. A one-dimensional equationOne might expect that the substitution (2.7) wouldlead to an asymptotic boundary layer equation involvingonly z derivatives of A in the limit d → 0, correspondingto the thermodynamic balance wTz ø ky Tzz. However,this does not happen. The horizontal advective terms oforder d21 that arise from the substitution (2.7) vanishidentically from (2.12), but the leading-order horizontaladvective terms that remain in (2.12) are still of order1, the same order as the leading-order vertical advectiveterm. Consequently, the thermodynamic balance doesnot, in general, reduce to wTz ø ky Tzz as ky → 0. Thismight be anticipated from the observation that the ver-tical velocity itself vanishes in this limit.Special solutions of (2.12) may still be sought inwhich horizontal advection does vanish, either identi-cally or to leading order. The matching conditions forA are themselves independent of x and y. In the case of(2.10), this reduction is possible because the Sverdruprelation enforces a proportionality between hx and wE/hat each point. This suggests the substitution A(x, y, z)5 B(z) in (2.12), or M(x, y, z) 5 d2T0B(z). If the re-sulting equation for B(z) were independent of x and y,then a one-dimensional boundary layer theory wouldexist in the two-layer limit. This substitution gives3 4dB d B d B22 22b f h T 5 k d . (3.1)x 0 y3 4dz dz dzComparison with (2.1) shows that this is a vertical ad-vective–diffusive balance in which the term Mx is re-placed by Mzhx, a consequence of fixing the isothermslopes to first order. The equality in (3.1) can in generalbe satisfied only if the quantity ky f 2/(bhxT0) is constantsince the latter is clearly independent of z while byassumption B depends only on z. From (2.5),2k f k hy y5 , (3.2)bh T wx 0 Ebut inspection shows that h/wE is not generally constant.Thus, an asymptotic one-dimensional internal boundarylayer theory for the two-layer limit of (2.1) does notexist in general. A solution may still exist that ap-proaches the two-layer solution in the limit ky → 0, and,if it exists, it must have d } , but the corresponding1/2kyinternal boundary layer will in general not be indepen-dent of horizontal position; that is, it will remain in-trinsically three-dimensional.If the expression (3.2) were constant, the equation(3.1) would reduce to the equation ultimately solved byYoung and Ierley (1986) in their asymptotic analysis ofthe similarity solution discovered by Stommel and Web-ster (1962). This can be seen as follows. Set1/2 1/22k f k hy yd 5 5 (3.3)1 2 1 2bh T wx 0 Eand letdBF(z) 5 . (3.4)dzThen (3.1) isFF0 5 F-, (3.5)and the boundary conditions (2.10) and (2.11) areF(z → 1`) → z, F9(z → 2`) → 0, (3.6)where prime denotes derivative with respect to the ar-gument. The two conditions (3.6) are sufficient for thethird-order equation (3.5) since the first also imples F9(z→ 1`) → 1. The abyssal upwelling velocity is deter-mined by the value c, where F(z → 2`) → c, sincew(z → 2`) → cdwE/h.The equation (3.5) with the boundary conditions (3.6)is solved by Young and Ierley (1986) in their analysisof the Stommel–Webster similarity solution (with thesign of z reversed). Their solution yields c 5 0.875 74.Note that the Stommel–Webster similarity solution isnot of ‘‘two-layer’’ type: it retains zonal temperaturegradients above and below the internal boundary layereven in the limit ky → 0. Near the internal boundarylayer, the two-layer model is a more accurate represen-tation of the numerical solutions of Samelson and Vallis(1997) than is the Stommel–Webster similarity solutionbecause of the large zonal temperature gradients in thelatter and the dependence of the thermocline depth inthe numerical solutions on horizontal position, whichroughly follows the two-layer solution.As an explicit example of a two-layer solution withthis type of internal boundary layer, consider (2.5) withH0 5 0 and wE 5 af 2(x 2 xE), where a is a constant.In this case, wE/h is constant, and (3.1) is independentof x and y and has the form (3.5), with boundary con-ditions (3.6). This is a two-layer solution with a one-dimensional internal boundary layer that has thicknessd } and is independent of horizontal position. In1/2kythis case, the thermodynamic balance in the boundarylayer does reduce to wTz ø ky Tzz.Although (3.5) is formally valid only if d from (3.3)is constant, as in the preceding example, it may stillprovide a useful approximation to M if d is only ap-proximately constant. If(dx, dy)/d ; d, (3.7)2102 VOLUME 29J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Ythen (3.5) will be the correct leading-order approxi-mation, and (3.7) will in turn be satisfied if|=(wE/h)|/(wE/h) ; d. (3.8)That is, the accuracy of the approximation will be con-trolled by the extent to which the vertical convergencewz in the two-layer solution is horizontally uniform. Thisrequires that the fractional variations in wE and h eitherbe separately small or cancel to first order. The condition(3.7) can be recast in terms of wE and H0 using (2.5).If wE is independent of x and y, and H0 is sufficientlylarge, then the approximation will be accurate. If wE isindependent of x, and h(x) 2 H0 ø H0, then the ap-proximation may be inaccurate.4. Related examplesSalmon and Hollerbach (1991) obtained some specialsolutions of the thermocline equations that are relevantto the present discussion. In one class of solutions (their‘‘S12’’), the temperature changed abruptly across an in-ternal boundary layer, as in the solution discussed above.In a second class of solutions (their ‘‘S13’’), the potentialvorticity changed abruptly across an internal boundarylayer, while the temperature field remained smooth asky → 0. For each of these classes, they presented specificexamples for the special case in which the boundarylayer is located at a constant depth, independent of hor-izontal 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 , as1/2kyin the solution discussed above.It is especially interesting that a boundary layer1/2kyappears also in their second example, for which the jumpis in potential vorticity rather than temperature. An ex-trapolation of the argument given above would seem tosuggest that the boundary layer thickness should be pro-portional to , as in the traditional advective–diffusive1/3kyscaling, since the potential vorticity fTz 5 fMzzz (ratherthan temperature T 5 Mzz) must be matched outside theboundary layer, and this would appear to lead to a factord3 in the scaling (2.7). In this case, however, M in theboundary layer may have contributions of zero, first,and second order in d, along with the third-order termassociated with the potential vorticity matching. Thus,the simple extrapolation is misleading, and the boundarylayer again scales with .1/2kyAcknowledgments. This research was supported bythe National Science Foundation, Division of OceanSciences (Grants OCE94-15512 and OCE98-96184). Iam grateful for comments from J. Pedlosky and R. Salm-on.REFERENCESLuyten, J., J. Pedlosky, and H. Stommel, 1983: The ventilated ther-mocline. J. Phys. Oceanogr., 13, 292–309.Parsons, A. T., 1969: A two-layer model of Gulf Stream separation.J. Fluid Mech., 39, 511–528.Pedlosky, J., 1979: Geophysical Fluid Dynamics. Springer-Verlag,624 pp.Salmon, R., 1990: The thermocline as an ‘‘internal boundary layer.’’J. Mar. Res., 48, 437–469., and R. Hollerbach, 1991: Similarity solutions of the thermoclineequations. J. Mar. Res., 49, 249–280.Samelson, R., and G. K. Vallis, 1997: Large-scale circulation withsmall diapycnal diffusion: The two-thermocline limit. J. Mar.Res., 55, 223–275.Stommel, H., and J. Webster, 1962: Some properties of the thermo-cline equations in a subtropical gyre. J. Mar. Res. 44, 695–711.Veronis, G., 1973: Model of the world ocean circulation, I: Wind-driven, two-layer. J. Mar. Res., 31, 228–288.Welander, P., 1971: The thermocline problem. Philos. Trans. Roy.Soc. London, Series A, 270, 415–421.Young, W. R., and G. Ierley, 1986: Eastern boundary conditions andweak solutions of the ideal thermocline equations. J. Phys.Oceanogr. 16, 1884–1900.

Internal Boundary Layer Scaling in “Two Layer” Solutions of the Thermocline Equations

https://ir.library.oregonstate.edu/downloads/z029p623d

Internal Boundary Layer Scaling in “Two Layer” Solutions of the Thermocline Equations

Abstract

Similar works

Full text

Available Versions

ScholarsArchive@OSU