is proportional to the amount by which this velocity is not parallel to the direction of the thermal wind shear v z = v v, (V_rotate_03)

Transcription

1 97 Whe globally itegrated over complete desity surfaces, the total trasport due to these o- liear processes ca be calculated. I gree is the mea diaeutral trasport from the ill- defied ature of eutral surfaces, blue is the diaeutral trasport due to cabbelig, red due to thermobaricity, ad black is the total global diaeutral trasport due to the sum of these three o- liear processes. We coclude from this that while the mea diaeutral trasport from the ill- defied ature of eutral surfaces is of leadig order locally, it spatially averages to a very small trasport over a complete desity surface. By cotrast, cabbelig ad thermobaricity are predomiatly dowwards advectio everywhere, so there is little such cacellatio o area itegratio with these processes.

2 Rotatio of the horiotal velocity with height Defie the agle ϕ (measured couter- clockwise) of the horiotal velocity v with respect to due east so that 98 v = v ( cosϕ, siϕ ). (V_rotate_01) Vertically differetiate this equatio ad take the cross product with v to obtai v v = kϕ v, (V_rotate_0) which shows that the rate of spiralig of the horiotal velocity vector i the vertical ϕ is proportioal to the amout by which this velocity is ot parallel to the directio of the thermal wid shear v. The last equatio ca be rewritte as ϕ v = k v v = uv vu = v k v = v v, (V_rotate_03) which demostrates that the rotatio of the horiotal velocity with height is proportioal to the helicity of the horiotal velocity, v v. Now, substitutig Eq. (3.1.3) for the thermal wid v, amely f v = 1 ρ ( ) k P + 1 ρ k P ito Eq. (V_rotate_03) we fid ( ) = g ρ k ρ = N ϕ v = v k v = N fgρ v P. Uder the usual Boussiesq approximatio gρ of the eutral taget plae,, so that we have k P, gρ (3.1.3) (V_rotate_04) ( ) 1 P is set equal to the slope ϕ v N f v, (V_rotate_05) ad sice the vertical velocity through geopotetials, w, is give by the simple geometrical equatio (where e is the vertical velocity through the eutral taget plae), we have w = t + v + e, ϕ v N f (V_rotate_06) ( w e t ), (V_rotate_07) showig that the rotatio of the horiotal velocity vector with height is ot simply proportioal to the vertical velocity of the flow but rather oly to the slidig motio alog the eutral taget plae, v.

3 The absolute velocity vector i the ocea 99 Neutral helicity is proportioal to the compoet of the vertical shear of the geostrophic velocity ( v, the thermal wid ) i the directio of the temperature gradiet alog the eutral taget plae, sice, from Eq. (3.1.3), amely f v = N k P, gρ ad the third lie of (3.13.), amely H = g 1 N T b ( P ) k, we fid that H = T fv (3.13.4) ρ b. This coectio betwee eutral helicity ad a aspect of the horiotal velocity vector motivates the idea that the mea velocity might be somehow liked to eutral helicity, ad this lik is established i this sectio. The absolute velocity vector i the ocea ca be writte as a closed expressio ivolvig the eutral helicity, ad this expressio is derived as follows. First the water- mass trasformatio equatio (A.3.1) is writte as v ( ) + KgN ( C b + Tb P) = γ γ 1 K + Dβ gn 3 d (3.13.7) Ŝ A d Ψ t, where the thickess- weighted mea velocity of desity- coordiate averagig, ˆv, has bee writte as ˆv = v + Ψ, that is, as the sum of the Euleria- mea horiotal velocity v ad the quasi- Stokes eddy- iduced horiotal velocity Ψ (McDougall ad McItosh (001)). The quasi- Stokes vector streamfuctio Ψ is usually expressed i terms of a imposed lateral diffusivity ad the slope of the locally- refereced potetial desity surface (Get et al., (1995)). More geerally, at least i a steady state whe t is ero, the right- had side of Eq. (3.13.7) is due oly to mixig processes ad oce the form of the lateral ad vertical diffusivities are kow, these terms are kow i terms of the ocea s hydrography. Eq. (3.13.9) is writte more compactly as v τ = v where τ, (3.13.8) ad v is iterpreted as beig due to mixig processes. The mea horiotal velocity v is ow split ito the compoets alog ad across the cotours of o the eutral taget plae so that v = v τ k + v τ, (3.13.9) where v = v τ k. Note that if τ poits orthwards the τ k poits eastward. The expressio v τ = v of Eq. (3.13.8) is ow vertically differetiated to obtai v τ = v τ + v = N k fgρ P τ + v, ( ) where we have used the thermal wid equatio (3.1.3), v = N k fgρ P. We will ow show that the left- had side of this equatio is φ v where φ is the rate of rotatio of the directio of the uit vector τ with respect to height (i radias per metre). By expressig the two- dimesioal uit vector τ i terms of the agle φ (measured couter- clockwise) of τ with respect to due orth so that τ = ( siφ, cosφ), we see that τ k = ( cosφ, siφ), τ = φ τ k ad k τ τ = φ. Iterestigly, φ is also equal to mius the helicity of τ (ad to mius the helicity of τ k ), that is, φ = τ τ = ( τ k) ( τ k), where the helicity of a vector is defied to be the scalar product of the vector with its curl. From the velocity decompositio (3.13.9) ad the equatio τ = φ τ k we see that the left- had side of Eq. ( ), v τ, is φ v, hece v ca be expressed as v = N fgρ k P τ φ v or v = φ H v, ( ) φ ρ f T b φ

4 where we have used the defiitio of eutral helicity H, Eq. (3.13.). The full expressio for both horiotal compoets of the mea horiotal velocity vector v is the 100 v = N k P τ fgρ φ v φ k + v. (3.13.1) Neutral helicity arises i this cotext because it is proportioal to the compoet of the thermal wid vector v i the directio across the cotour o the eutral taget plae (see Eq. (3.13.4)). This equatio (3.13.1) for the Euleria- mea horiotal velocity v shows that i the absece of mixig processes (so that v = v = 0 ) ad so log as (i) the epieutral cotours do spiral i the vertical ad (ii) is ot ero, the eutral helicity H (which is proportioal to k P τ ) is required to be o- ero i the ocea wheever the ocea is ot motioless. Plaetary potetial vorticity Plaetary potetial vorticity is the Coriolis parameter f times the vertical gradiet of a suitable variable. Potetial desity is sometimes used for that variable but usig potetial desity (i) ivolves a iaccurate separatio betwee lateral ad diapycal advectio because potetial desity surfaces are ot a good approximatio to eutral taget plaes ad (ii) icurs the o- coservative barocliic productio term of Eq. (3.13.4). Usig approximately eutral surfaces, as, (such as Neutral Desity surfaces) provides a optimal separatio betwee the effects of lateral ad diapycal mixig i the potetial vorticity equatio. I this case the potetial vorticity variable is proportioal to the reciprocal of the thickess betwee a pair of closely spaced approximately eutral surfaces. The evolutio equatio for plaetary potetial vorticity is derived by first takig the epieutral divergece of the geostrophic relatioship from Eq. (3.1.1), amely fv = g k. The projected divergeces of a two- dimesioal vector a i the eutral taget plae ad i a isobaric surface, are a = a + a ad a = a + a from which we fid (usig Eq. (3.1.6), = P P ) a = a + a P P. (3.0.1) Applyig this relatioship to the two- dimesioal vector have fv = g k we ( fv) = g ( k ) + fv P P = 0. (3.0.) The first part of this expressio ca be see to be ero by simply calculatig its compoets, ad the secod part is ero because the thermal wid vector v poits i the directio k P (see Eq. (3.1.3)). It ca be show that r fv ( ) = 0 i ay surface r which cotais the lie P ρ. Eq. (3.0.), amely ( fv) = 0, ca be iterpreted as the divergece form of the evolutio equatio of plaetary potetial vorticity sice fv ( ) = q v = 0, (3.0.3) γ where q = f γ is the plaetary potetial vorticity, beig the Coriolis parameter times the vertical gradiet of Neutral Desity. This istataeous equatio ca be averaged i a thickess- weighted sese i desity coordiates yieldig ˆq ˆv γ = v q = γ ( γ ), (3.0.4) 1 K ˆq

5 where the double- primed quatities are deviatios of the istataeous values from the thickess- weighted mea quatities. Here the epieutral eddy flux of plaetary potetial vorticity per uit area has bee take to be dow the epieutral gradiet of ˆq with the epieutral diffusivity K. The thickess- weighted mea plaetary potetial vorticity is ˆq γ q γ γ 101 = f γ, (3.0.5) ad the averagig i the above equatios is cosistet with the differece betwee the thickess- weighted mea velocity ad the velocity averaged o the Neutral Desity surface, ˆv v (the bolus velocity), beig ˆv v = K l( ˆq ), sice Eq. (3.0.4) ca be writte as ( f ˆv ) = ( γ 1 K ˆq ) while the average of Eq. (3.0.3) is f v ( ) = 0. The divergece form of the mea plaetary potetial vorticity evolutio equatio, Eq. (3.0.4), is quite differet to that of a ormal coservative variable such as Absolute Saliity or Coservative Temperature, γ t ˆv + γ + ( e ) = γ γ 1 K ( ( ) + D ), ( γ _Eq.) because i Eq. (3.0.4) the followig three terms are missig; (i) the vertical diffusio of ˆq with diffusivity D (ii) the diaeutral advectio of ˆq by the diaeutral velocity e, ad (iii) the temporal tedecy term. The mea plaetary potetial vorticity equatio (3.0.4) may be put ito the advective form by subtractig ˆq times the mea cotiuity equatio, 1 γ from Eq. (3.0.4), yieldig or t + ˆv γ ˆq + ˆv ˆq t = γ 1 K ˆq ˆq t + ˆv ˆq + e ˆq = d ˆq dt + e = 0, (3.0.6) γ ( γ ) + ˆqe, (3.0.7) ( ) + ˆqe = γ γ 1 K ˆq ( ). (3.0.8) I this form, it is clear that potetial vorticity behaves like a coservative variable as far as epieutral mixig is cocered, but it is quite ulike a ormal coservative variable as far as vertical mixig is cocered; cotrast Eq. (3.0.8) with the coservatio equatio for Coservative Temperature, t + ˆv + e = d dt = γ ( γ 1 K ) + ( D ). (A.1.15) If ˆq were a ormal coservative variable the last term i Eq. (3.0.8) would be ( D ˆq ) where D is the vertical diffusivity. The term that actually appears i Eq. (3.0.8), ( ˆqe ), is differet to ( D ˆq ) by ( ˆqe D ˆq ) = f ( e γ Dγ ). Equatio (A..4) for the mea diaeutral velocity e ca be expressed as e D + Dγ γ if the followig three aspects of the o- liear equatio of state are igored; (1) cabbelig ad thermobaricity, () the vertical variatio of the thermal expasio coefficiet ad the salie cotractio coefficiet, ad (3) the vertical variatio of the itegratig factor b( x, y,) of Eqs. (3.0.10) - (3.0.15) below. Eve whe igorig these three differet implicatios of the oliear equatio of state, the evolutio equatios (3.0.7) ad (3.0.8) of ˆq are ulike ormal coservatio equatios because of the extra term ( ) = f ( e γ D ) f ( D ) = ( D ˆq ) (3.0.9) ˆqe D ˆq γ γ

6 o their right- had sides. This presece of this additioal term ca result i umixig of ˆq i the vertical. Cosider a situatio where both ˆq ad are locally liear fuctios of ŜA dow a vertical water colum, so that the ŜA ˆq ad Ŝ A diagrams are both locally straight lies, exhibitig o curvature. Imposig a large amout of vertical mixig at oe height (e. g. a delta fuctio of D ) will ot chage the ŜA diagram because of the ero ŜA curvature (see the water- mass trasformatio equatio (A.3.1)). However, the additioal term ( D ˆq ) of Eq. (3.0.9) meas that there will be a chage i ˆq of ( D ˆq ) = ˆqD + ˆq D ˆqD. This is ˆq times a egative aomaly at the cetral height of the extra vertical diffusio, ad is ˆq times a positive aomaly o the flakig heights above ad below the cetral height. I this way, a delta fuctio of extra vertical diffusio iduces structure i the iitially straight Ŝ A ˆq lie which is a telltale sig of umixig. This plaetary potetial vorticity variable, ˆq = f γ, is ofte mapped o Neutral Desity surfaces to give isight ito the mea circulatio of the ocea o desity surfaces. The reasoig is that if the ifluece of diaeutral advectio (the last term i Eq. (3.0.7)) is small, ad the epieutral mixig of ˆq is also small, the i a steady ocea ˆv ˆq = 0 ad the thickess- weighted mea flow o desity surfaces ˆv will be alog cotours of thickess- weighted plaetary potetial vorticity ˆq = f γ. Because the square of the buoyacy frequecy, N, accurately represets the vertical static stability of a water colum, there is a strog urge to regard fn as the appropriate plaetary potetial vorticity variable, ad to map its cotours o Neutral Desity surfaces. This urge must be resisted, as spatial maps of fn are sigificatly differet to those of ˆq = f γ. To see why this is the case the relatioship betwee the epieutral gradiets of ˆq ad fn will be derived. For the preset purposes Neutral Helicity will be assumed sufficietly small that the existece of eutral surfaces is a good approximatio, ad we seek the itegratig factor b = b( x, y,) which allows the costructio of Neutral Desity surfaces (γ surfaces) accordig to γ = b β S γ A α ( ) = b ρ ρ κ P. (3.0.10) Takig the curl of this equatio gives b b 10 ρ κ P ρ = κ P. (3.0.11) The bracket o the left- had side is ormal to the eutral taget plae ad poits i the directio = + k ad is g 1 N ( + k). Takig the compoet of Eq. (3.0.11) i the directio of the ormal to the eutral taget plae,, we fid 0 = κ P = ( κ + κ ) ( P + P ) ( ) P k = κ P = κ P k = κ SA S A + κ = T b P k = g N H, (3.0.1) which simply says that the eutral helicity H must be ero i order for the diaeutral compoet of Eq. (3.0.11) to hold, that is, P k must be ero. Here the equalities κ SA = β P ad κ = α P have bee used. Sice b ca be writte as b = b + b, Eq. (3.0.11) becomes ( ) = P κ ( + k), (3.0.13) ( + k) has bee used o the right- had side, ( + k) g 1 N lb + k where P = P beig the ormal to the isobaric surface. Cocetratig o the horiotal

7 compoets of this equatio, g 1 N lb = P κ, ad usig the hydrostatic equatio P = gρ gives lb = ρg N κ = ρg N α P β P S A 103 ( ). (3.0.14) The itegratig factor b defied by Eq. (3.0.10), that is b ( ρ l γ ) γ ρ l ( ρ l ρ l ) where ρ l ρ l (β S A α ), allows spatial itegrals of b (β S A α ) = b l ρ l lγ to be approximately idepedet of path for vertical paths, that is, for paths i surfaces whose ormal has ero vertical compoet. By aalogy with fn, the Neutral Surface Potetial Vorticity ( NSPV ) is defied as gγ 1 times ˆq = f γ, so that NSPV = b fn (havig used the vertical compoet of Eq. (3.0.9)), so that the ratio of NSPV to fn is foud from Eq. (3.0.14) to be NSPV = b = ρl γ fn l γ ρ { as ( ) dl} { dl as }. = exp ρg N α P β P S A = exp ρg N κ (3.0.15) The itegral here is take alog a approximately eutral surface (such a Neutral Desity surface) from a locatio where NSPV is equal to fn. The deficiecies of fn as a form of plaetary potetial vorticity have ot bee widely appreciated. Eve i a lake, the use of fn as plaetary potetial vorticity is iaccurate sice the right- had side of (3.0.14) is the ρg N α P = ρg N α P P P = α P α P, (3.0.16) where the geometrical relatioship = P P has bee used alog with the hydrostatic equatio. The mere fact that the Coservative Temperature surfaces i the lake have a slope (i. e. P 0 ) meas that the spatial variatio of cotours of fn will ot be the same as for the cotours of NSPV i a lake.

8 104 I the situatio where there is o gradiet of Coservative Temperature alog a Neutral Desity surface ( γ = 0 ) the cotours of NSPV alog the Neutral Desity surface coicide with those of isopycal- potetial- vorticity ( IPV ), the potetial vorticity defied with respect to the vertical gradiet of 1 potetial desity by IPV = fgρ ρ. IPV is related to fn by (McDougall (1988)) ( pr) ( ) ( pr) ( ) 1 R r 1 IPV gρ ρ β ρ β 1 1 = =, (3.0.17) fn N β p Rρ 1 β p G G so that the ratio of NSPV to IPV plotted o a approximately eutral surface is give by ( p) ( ) NSPV β Rρ 1 = exp { g N ( ) as P ρκ d }. l (3.0.18) IPV β pr Rρ r 1 The sketch below idicates why NSPV is differet to IPV ; it is because of the highly differetiated ature of potetial vorticity that isolies of IPV ad NSPV do ot coicide eve at the referece pressure p r of the potetial desity 3 variable. NSPV, fn ad IPV have the uits s.