Neutral-surface potential vorticity

Prog. Oceanog. Vol. 20, pp. 185-221, 1988.

0079-6611/88 $0.00 + .50 Copyright ~) 1989 Pergamon Press plc

Printed in Great Britain. All rights reserved.

Neutral-Surface Potential Vorticity TREVOR J. McDoUGALL CS1RO Division of Oceanography, GPO Box 1538, Hobart, TAS 7001, Australia (Submitted September 1988;Accepted October1988)

ABSTRACT

The use of potential density in ocean models does not overcome all

the annoying problems

caused by the compressible nature of seawater. For example, the reduced gravity for layered ocean models is not proportional to the difference of potential density across an interface. The appropriate reduced gravity is derived and i t

is shown to be r e l a t i v e l y easy to incorp-

orate in future layered models. Potential-density surfaces are commonly inclined with respect to neutral surfaces and so lateral motion along neutral surfaces gives rise to unwanted diapycnal velocities that are unrelated to vertical mixing processes.

This unphysical part

of the diapycnal motion is quantified, emphasizing the need for performing ocean-circulation studies in the neutral-surface framework. I f there were no variations of 0 and S along neutral surfaces in the ocean, then lateral mixing would occur along potential-density surfaces.

A perfectly respectable potential vort-

i c i t y variable would then be isopycnal potential v o r t i c i t y , IPV, defined by - g f ~ p ~ / @ z ] ~ . However, there is mixing in the ocean, and neutral surfaces do not coincide with potential density surfaces.

Since the e f f i c i e n t lateral mixing processes occur along neutral trajec-

tories, and vertical velocities through neutral surfaces occur only in response to mixing processes of a vertical nature (including the two dianeutra] advection processes, thermob a r i c i t y and cabbeling), i t is logical to seek a potential v o r t i c i t y variable that is proportional to f / h ,

where h is the vertical distance between adjacent neutral surfaces.

This

form of potential v o r t i c i t y is here referred to as "neutral-surface potential v o r t i c i t y " , NSPV.

Its epineutral variations can be s i g n i f i c a n t l y different to those of the two other

commonly used potential v o r t i c i t y variables, IPV, and fN 2.

The difference between NSPV and

fN 2 is shown to result from the path-dependence inherent in integrating N2 in the vertical plane between two neutral trajectories. The B-spiral equations are derived for a f u l l y non-linear equation of state, surfaces emerge as the dynamically relevant surfaces for these studies.

It

and neutral

is also shown

that IPV is not conserved on potential-density surfaces because of the non-linear nature of the equation of state; this problem does not arise in the neutral tangent plane.

Mixing

processes cause scalar properties to change at different rates on a potential-density surface 2o:~-c 185

JPo

186

T.J. McDoUGALL

than in a neutral tangent plane.

In particular, potential density i t s e l f is mixed and advec-

ted by epineutral processes, thereby complicating the interpretation of inverse models that use potential density surfaces.

The dynamical equations governing the ocean circulation

take simple and exact forms with respect to the neutral tangent plane at eachpoint, but when an interface or surface must be formed over an extended geographical region, the ambiguity associated with defining a neutral surface raises i t s ugly head. Here i t is shown that the approximately neutral surfaces of McDOUGALLand JACKETT (1988) are the most appropriate surfaces for this purpose.

CONTENTS 1.

Introduction

187

2.

Reduced gravity in layered ocean models

189

3.

Vertical, diapycnal and dianeutral velocities

191

4.

The thermal wind equation

194

5.

The continuity equation

195

6.

Neutral-surface potential v o r t i c i t y

196

7.

The variation of hN2 along neutral trajectories: two derivations

197

7.1

An expression~ for hiN~/hN2^ by using Stokes's theorem

198

7.2

Finding hiN~/hN2 by keeping track of potential density

200

8.

NSPV compared with IPV and fN 2

203

9.

The beta-spiral method

204

10.

Conservation statements in potential-density surfaces

207

10.1

Nonconservation of IPV in potential-density surfaces

207

10.2

A geometrical constraint for the conservation of potential v o r t i c i t y

209

10.3

Conservation equations for scalars in potential-density surfaces

211

11.

Approximately neutral surfaces and layered models

12.

Conclusions

212 216

13.

Acknowledgements

218

14.

List of Symbols

218

15.

References

220

Neutral-surface potential vorticity

1.

187

INTIIODUCTIOII

In a previous paper, (McDOUGALL, 1987a), some basic properties of neutral surfaces derived and three neutral surfaces from the North Atlantic were mapped.

were

In the previous

paper in this issue of the journal, (McDOUGALLand JACKETT, 1988), the ambiguity involved in defining a neutral surface was addressed. This ambiguity (or path-dependence) was shown to cause vertical motion by the lateral movement of f l u i d along neutral trajectories that are actually helices or spirals in space.

Epineutral mixing and advection then result in

mean vertical advection without requiring vertical mixing processes to work against gravity. This new type of mean vertical advection in the ocean remains to be quantified.

Because

of the i l l - d e f i n e d nature of a neutral surface, any mathematically well-defined surface in space w i l l not quite have the "neutral property" that f l u i d parcels can be moved small distances in the surface without experiencing a gravitational restoring force.

This d i f f i c u l t y

does not arise when considering property gradients and conservation equations since path-dependence increases quadratically with lateral distance.

at a p o i n t ,

Hence, neutral tangent

planes are well defined while neutral surfaces are not. The present paper develops the dynamical equations that govern the general circulation of the ocean in the framework of the neutral tangent plane at each point in space, under the hydrostatic, geostrophic

and Boussinesq approximations.

S i n c e the conservation equations

are written in terms of spatial and temporal gradients at a point, the path-dependent nature of neutral surfaces does not arise.

In section 11 an application is explored where a well-

defined surface is required over a complete ocean basin; i t is shown that the approximately neutral surfaces of McDOUGALL and JACKETT (1988) are sufficiently accurate for that purpose. In the remainder of the introduction, some of the properties of neutral surfaces are reviewed (from McDOUGALL, 1987a), followed by a guide to the rest of the paper. Lateral mixing in the ocean occurs along neutral trajectories along which variations of potent i a l temperature compensate the changes of s a l i n i t y , through the ratio of the saline contraction coefficient, B, and the thermal expansion coefficient, ~, defined by

::

_L

where p is the

P

S,p

;

B=~

O,p

,

i n s i t u density of seawater (McDOUGALL, 1987a).

(I) I f Vn is the projected two-

dimensional gradient operator in a neutral tangent plane, (see the l i s t of symbols for the exact definitions of ~, B and Vn) we have ~ n 0 = B~nS, where ~ and B are evaluated at the i n s i t u p r e s s u r e , p.

By contrast, the variations of 0 and S in a potential-density surface

are related by ~(pr)~oO = B(Pr)VoS where ~ and B are evaluated at the chosen reference pressure, Pr" Whereas a f l u i d parcel can be moved small distances along a neutral trajectory without experiencing any buoyant restoring force, the same is not true for motion in a potent i a l - d e n s i t y surface.

These different surfaces diverge in the v e r t i c a l , and so the spatial

gradients of potential temperature in the two surfaces are different, being related by

188

T . J . McDOUOALL

C[Rp- I ]

~e:

~

~ne:

~n e

,

(2)

where Rp, the s t a b i l i t y ratio of the water column, and c are defined by

Rp = aBz/BSz

;

e(p)/B(p) c = e(pr)/B(Pr )

(3)

The factor ~ has been introduced as a convenient shorthand notation for C[Rp- 1]/[Rp- c]. The ratio, c, of the expansion coefficients at p to those at a fixed reference pressure, Pr' is about 1.2 at a pressure difference, [p-pr ] of 1000db, and for B = 4°C. Taking a typical value of Rp of 5, we see from (2) that ~ = 1.26 and that ~a8 is typically 26% larger than 9nB for [p-pr] ~ 1000db. Inverse models that use the lateral gradients of properties to determine the strengths of various mixing processes w i l l obviously yield different results depending on the surfaces in which the lateral gradients of properties are evaluated. The aim of this paper is to derive the dynamical equations that apply to the ocean circulation, paying careful attention to the complicated nature of the equation of state of seawater. Neutral tangent planes emerge as the natural reference frame for dynamical ocean studies, and the errors made by using potential-density surfaces are quantified.

Since lateral mixing

and movement of water-parcels occurs along neutral trajectories (epineutral mixing and advection), i t is logical to take approximately neutral surfaces to be the interfaces in layered ocean models. The vertical velocity component throughthe interfaces is then directly interpretable in terms of mixing processes (see sections 10 and 11).

Similarly, the most approp-

riate surface in which to evaluate and plot f l u i d properties (such as potential v o r t i c i t y ) is the approximately neutral surface. In section 2 of this paper, the reduced gravity that should be used in layered models is related to the normal reduced gravity that is based on differences of potential density across an interface.

A very simple relationship is found that w i l l allow the correct buoyant driving

force to be used in the thermal wind equation of future layered circulation models. Section 3 discusses the differences between vertical, diapycnal and dianeutral velocities.

I t is

shown that lateral motion along a neutral surface (epineutral motion) may involve a large diap~cnal velocity that is not caused by mixing processes. The vertical shear of the lateral

current (thermal wind, section 4) is driven by the slope of the neutral tangent plane to the horizontal; this slope can be quite different (both in magnitude and direction) to the corresponding slope of a potential-density surface.

The continuity equation (section 5)

leads to the conservation equation of linear v o r t i c i t y with respect to neutral tangent planes, neutral-surface potential v o r t i c i t y (NSPV, section 6), defined to be proportional to the Coriolis frequency divided by the height between adjacent neutral tangent planes. Neutralsurface potential v o r t i c i t y is not simply proportional to fN 2 (where N2 is the square of the buoyancy frequency), because N2 displays path-dependence under integration around a vertical loop in the ocean (sections 7 and 8).

The beta-spiral method of determining the absolute

velocity vector is considered in section 9.

Again the neutral tangent planes emerge as the

dynamically relevant surfaces in the ocean. The nonlinear nature of the equation of state of seawater ensures that isopycnal potential v o r t i c i t y (IPV) is not conserved along the mean

Neutral-surface potential vorticity

189

flow direction in potential-density surfaces, even in the absence of diapycnal velocities and f r i c t i o n a l forces (section 10).

S i n c e existing inverse models use potential-density

surfaces to bound boxes (in inverse box models) or in the beta-spiral method, conservation equations for scalars are derived with respect to potential-density surfaces in section 10.2. I t is found that potential density is mixed not only by vertical mixing processes but also by epineutral mixing.

Finally, in section 11, layered ocean models are considered with the

interfaces being approximately neutral surfaces.

It

is shown that the errors associated

with these surfaces not being quite "neutral" are small, the difference between these surfaces and potential-density surfaces being far more important.

2.

REDUCEDGRAVITY IN LMERED OCEAN MODELS

Layered circulation models assume that the s t r a t i f i e d ocean can be approximated by a series of well-mixed layers, separated by sharp interfaces.

The best known recent example of such

models is the so-called "ventilated thermocline" model of LUYTEN, PEDLOSKY and STOMMEL(1983), the physics of which is explained very well in LUYTEN and STOMMEL (1986).

The potential

v o r t i c i t y of each layer in such models is assumed to be constant along mean streamlines, except for the vortex stretching term resulting from water flowing across the interfaces. Here the appropdateversion of the thermal wind equation is examined for layered models. Figure I shows a cross-section through several

layers and interfaces of a layered model.

For the purpose of deducing the relevant reduced gravity in such a model, the nature of the interfaces is not important (i.e.

whether they are neutral surfaces or potential-density

surfaces or indeed any other kind of interface).

The geostrophic relations, -fv = -px/p and

fu = -py/P, apply in each layer, and the hydrostatic balance, Pz = -gP' holds in the v e r t i c a l , implying that the thermal wind equations expressing the differences in the lateral velocity components between the layers are (using the Boussinesq approximation)

Au :

-

and

Av :

T

'

where Zy and xx are the slopes of the interface, defined by z = z ( x , y ) , to the in the y and x directions.

(4)

geopotential

The important point to note in (4) is that'the relevant reduced

gravity that appears in the thermal wind equation of layered models is not that based on potential density, g APo/PB' but is that based on the in s i t u density difference, at the interface.

g Ap/p,

S i n c e the layers are v e r t i c a l l y well-mixed, potential temperature and

s a l i n i t y do not vary with depth within each layer, and at the interface between a pair of layers there are jumps of AB and AS.

The difference of in s i t u density, Ap, across an inter-

face is given by =--~-~= BAS - ~Ae + YAp P

,

where B, a and Y are evaluated at the i n s i t u v a l u e s of S, O and p at the interface. trast, the change of potential density across the interface, APO, is

(5) By con-

190

T . J . McDouOALL

Z'

B,S,p,pe _

---

Ap

interfaces

e

vertical cast

FIG.I.

s

Pe

P

P

A vertical crossection through a layered ocean. A vertical cast shows that each layer is well-mixed in e, s and potential density, Pe ; however, the in s i t u density, p, increases with pressure in each layer because of the compressibility of seawater. At the reference pressure, Pr (taken to be Pr = Odb), PB=P" The density difference that enters the thermal wind equation is Ap, which is larger than ApB by the factor p = c[R o - I]/[Ro - c] (see equation 7). Note that a layered model does not requ3re that 8 and S be independent of lateral position along each layer.

APB PB - B(pr) AS - ~(pr)AB

(6)

At the interface, Ap is zero and so the ratio of ap to Ape at the interface is given by Ap/p

Ap-~o

= ~-T __

C[Rp- I ] ~

~

C[Rp- I ] ~

:

~

(7)

,

where Rp = ~AB/~AS. The variations of B with pressure are much less than the corresponding variations of a, and so B(p)/B(pr) is approximately I. Usually layered ocean models assume potential density to be conserved in each layer so that there is a constant potential density difference across each interface.

Equation (4) shows,

however, that the relevant reduced gravity involves the difference of i n s i t u from (7) i t can be seen that the two reduced gravities d i f f e r by the factor u.

density, and This means

that the reduced gravity that acts at an interface must be allowed to vary with position as R changes and as the depth of the interface varies. Note that this multiplicative factor P p is the same factor that relates the lateral gradients of potential temperature in potentialdensity surfaces to those in a neutral surface (equation 2) and is also the same factor relating N2 to the vertical gradient of potential density ( i . e . relating fN 2 to IPV, equation 25 below).

This factor can be equal to 2 in the Central Water and is even larger in special

regions, such as under the Mediterranean Water outflow in the North Atlantic where R appP roaches c (implying that - @pe/@z ÷ 0), under the poleward extension of Western Boundary Currents (GORDON, 1981) or at the base of over-wintered Warm-Core Rings (SCHMITT and OLSON, 1985).

Neutral-surface potential vorticity

191

A recent example of the ventilated thermocline model is that for the South Pacific by DE SZOEKE (1987). PB = 27.30 and

His two deepest interfaces were chosen as the potential-density surfaces P0 = 26.90, with average pressures of 1100db and 500db respectively.

The

factor ~ is approximately 1.3 for the deeper interface and 1.2 for the shallower one, implying that the driving forces for all but the shallowest layer of his three and a half layer model were underestimated by between 20% and 30%.

Also, since

u

varies with lateral position

(because both c and Rp vary along an interface), the use of a reduced gravity based on potential

density w i l l miss these lateral variations in the geostrophic driving force.

I t is

now clear that Rp is no longer solely relevant to double-diffusive mixing studies, but that i t is also important in dynamical oceanography, through i t s appearance in u = C[Rp - 1]/[Rp-C]. While the most rational choice for an interface in a layered model is an approximately neutral surface, and this is the concern of the rest of this paper, the derivation of the appropriate reduced gravity in this section is independent of the neutral-surface framework. That is, even i f one took potential-density surfaces as the interfaces in a layered model (which would in fact be correct i f ~ 0 =~0), the correct reduced gravity is s t i l l related to the difference of potential density by (7).

The factor u differs from I in proportion to the pressure excur-

sion from one's reference pressure, Pr' and also depends on potential temperature.

I t is

quite straightforward then to build in the correct reduced gravity into future layered ocean models by parameterizing the variation of ~ as a function of position in the model. Perhaps the simplest way to evaluate u in an approximate fashion is to calculate a(pr ), which is a function only of O and S, from an equation of state of one's choice, and then to use (33) below to obtain c.

Then U follows from a knowledge of Rp . A further expression is derived in section 11 below for the appropriate reduced gravity of layered models (see equation 87),

using the results of section 7.

3.

VERTICAL, DIAPYCNAL AND DIANEUTRAL VELOCITIES

Mixing processes (especially vertical mixing) cause f l u i d to move through neutral surfaces. These motions can be modelled as vortex-stretching terms in layered models of ocean circulation.

However, since potential-density surfaces are commonly inclined with respect to neutral

surfaces, even the lateral movement along neutral surfaces motion through isopyonals

implies that there is vertical

(see Figure 2).

Let the height of a neutral tangent plane be given by z = N[x,y], where x and y are the distances in the eastward and northward directions. is expressed s i m i l a r l y as z = R[x,y].

The height of a potential-density surface

Neutral surfaces and potential-density surfaces inter-

sect along potential isotherms (see McDOUGALL, 1987a) so that the two-dimensional vector representing the difference, [VoR - Vn~], of the vector angles between the two surfaces and the geopotential, is parallel to Vno and V 0.

An expression for [~OR - ~n N] can be found

by realizing that the difference between the lateral gradients of potential temperature in a potential-density surface and in a neutral tangent plane, [Vo0

- V~],

is equal to the

vertical gradient of potential temperature, 0z, multiplied by [VOR- ~nN] so that

192

T.J.M c D O U G A L L

I'

I

wd ~ w d - el

w e

I [w- e]

~

= const.

isobar,z = P[x,y] FIG.2.

A vertical section containing the f l u i d ' s velocity vector. The vertical distances shown at the right of this figure are labelled by their respective vertical velocities and should all be multiplied by a small time interval, ~t, to give the vertical distances shown. Note that even i f there were no mixing (and therefore the dianeutral velocity, e, were zero), a lateral movent in the neutral surface implies the existence of a diapycnal velocity, , of -~n.~ne[~ - 1]/ez (see equation 11). The angle between a potentialdensity surface and a neutral tangent plane, [~oR - ~n#], is [~ - I~ times the angle between a neutral tangent plane and a potential isotherm, ~n 0/e z (see equation 8). The vertical velocity through a p~tential isotherm, we, is shown, and is given by we = e + #n.#n0/ez,so that [w - el = -[p - 1][w e - el. Also, the angle between a neutral surface and an isohaline, VnS/Sz, is R_ times that between a neutral tangent plane and a potential isotherm, Vn0/Bz. ~

~

Vn8 [VOR- ~n~]

-

~8

Bz [~ - I ]

1

]

8z [ I - ~ ]

(8)

J Taking Rp = 1.9 and c = 1.3, appropriate to [p - p~] ~ 1000 db, [~ - I ] is approximately I. Typical values of l#nel and ~ are 0.5 x I0-6*C m-1 and 0.5 x 10-2*C m"I respectively, which gives a typical slope between neutral surfaces and potential density ( i . e . isopycnal) surfaces of 10-4 . A typical lateral velocity of I0-2m s-I along a neutral surface w i l l then have a diapycnal component, ~n

[VnN - VoR], of about 10-7 m s- I , where Vn.qne has been taken

to be one tenth of IVnl IVn8I. neutral component.

This diapycnal velocity occurs even though there is no dia-

The difference between the slopes, VoR and ~/n~, expressed in (8) apply also when comparing the slopes of interfaces in layered ocean models that have either potential-density surfaces or neutral surfaces as the interfaces between the models' layers.

Since the interfacial

slopes are used in the thermal wind equation to predict the vertical shear of horizontal velocity, i t is clear that the use of different surfaces as interfaces in these models w i l l directly affect the vertical shear. I t is even possible for qoR to have a different sign to VnN, by which I mean that ~.VOR < 0, (see McDOUGALL, 1987a for an example), so implying a vertical shear of different sign using the two different coordinate systems.

I t is

Neutral-surface potential vorticity

193

emphasised that this effect is quite separate to the issue raised in the previous section of the paper concerning the appropria~ reduced gravity to use in such layered models, since the two forms of reduced gravity are different even when V~ = 0 and potential-density surfaces coincide with neutral surfaces. The cause of MULLER and WILLEBRAND's (1986) S-B mode of the ocean circulation can be understood from equation (8). Imagine an ocean with uniform upwelling, in which there is a gradient of potential temperature in potential-density surfaces, V~B, and the slope of a particular potential-density surface is ~oR. As water is slowly upwelled, ~aR and ~B w i l l remain constant apart from the effects of mixing processes, but ~ w i l l change because of the change in pressure and so the dynamically important slope of the neutral tangent plane, ~nN' must change (from 8). This dynamical mode of response of the ocean acts on timescales as long as 1000 years, but as yet we have no idea of i t s importance. Mixing processes are thought to lead to a dianeutral upwelling velocity of order I0-7m s-I (~ 3m yr - I ) in the deep ocean (MUNK, 1966), and so the aiapyonal velocities arising from considering oceanic processes relative to isop~cnals rather than to neutral surfaces can be as large as the physically relevant

dianeutral

velocities caused by mixing processes.

This may be one of the reasons why inverse models that use observed isopycnal surfaces for the interfaces between the model's layers (or boxes) are often not able to reach s t a t i s t i c a l l y significant conclusions about vertical mixing processes, since the vertical advection across the model's interfaces may be dominated by a component that is simply caused by resolving the lateral oceanic velocity along and through a surface that has no exact dynamical s i g n i f i cance; namely, a potential-density surface. LUYTEN, STOMMELand WUNSCH (1985) have performed a two-and-a-half layer ( i . e . the third layer was stationary) diagnostic study of a 10° by 5° box of the North Atlantic centred at 55°N, 22.5°W.

Their interfaces were potential-density surfaces and the deeper interface had an

average pressure of 600db.

I have mapped the neutral surface with the same average pressure

in this area and find that the eastward slope of the interface (Dx in their terminology) is 25% less than the eastward slope of the potential-density surface.

In addition, their

study used reduced gravities based on differences of potential densities; the correct reduced gravity is larger by the factor ~ = C[Rp- 1]/[Rp- c], which is about 1.3 at this depth.

For-

tuitously, the use of an inappropriate interface (a potential-density surface) in their model was almost compensated for by using the wrong reduced gravity!

This is purely coincidental

since ~ depends only on the pressure excursion from one's reference pressure and on the stabi l i t y ratio, Rp, whereas the difference between the slopes of neutral surfaces and potentialdensity surfaces, [ VOR- VnN], involves an additional piece of information; namely ~nB/Bz, the slope of a potential isothermal surface to the neutral surface. The vertical velocity through geopotentials, w, is of crucial importance to the general c i r culation of the ocean through the conserv~ion of potential v o r t i c i t y , and here w is compared to the dianeutral velocity, e, and the diapycnal velocity, wd (see Figure 2). The geometrical relations between these different types of vertical velocity are listed here for an ocean in steady-state.

194

T . I . McDouOALL

w = e * Vn.VnN

(9)

W= wd + Vn.VOR

(10)

wd = e + ~nV . lYnN~ - VoR] = e - Vn.VnO[U~ - I ] / 0 z

(11)

Note that both e and wd are actually vertical velocities aligned in the direction of the Cartesian unit vector k (see Figure 2).

I t is not, therefore, s t r i c t l y correct to call them

dianeutral or diapycnal velocities as these are normal to their corresponding surfaces. For example, the true diapycnal velocity is smaller than our wd by the factor [ I - R2 - R210"5 x y" ' which is less than unity by about 0.5 x 10-8 . I t is however, convenient to refer to e and wd as the dianeutral and diapycnal velocities, even though they are s t r i c t l y vertical velocity components with a s l i g h t l y greater magnitude than the diapycna] or dianeutral velocities.

4.

THE THERNAL WIND EQUATION

The geostrophic equations, -fv = -px/p

and fu = -py/p are differentiated with respect to

z, assuming the hydrostatic relation Pz = -gP' obtaining -fVz = -g[eex - BSx] + g[YPz - Pz/P]Px/Pz

'

and a similar relation for fu z in terms of y derivatives.

(12) Here px/p has been written as

BSx - a0x + YPx" Now g[aBx -BSx] is simply the i component of g [ ~ 0 - BVS] so that -g[a0 x - 6Sx] = -g[aV0 - BVS].~ = -N2 m.i

= N2 Nx

,

(13)

using the fact that eV% - 6VS is normal to a neutral tangent plane and is equal to N2m, where = [4Vx,-Ny,1]; Nx and Ny being the slopes of the neutral tangent plane in the x and y directions (see McDOUGALL and JACKETT, 1988). The second term (12) can be simplified by realizing that g[YPz

q / p ] is equal to N2, while px/pz is equal to -Px, Where z = P [ x , y ] is the height

of a surface of constant pressure above a geopotential, so that

-Vz =

~2

N2

[Nx - Px]

=

- g-t~p ~

I

,

(14)

n

and s i m i l a r l y ,

~z uz=

N2 a__e..I [~-

~]

=

~ - [Vn,V

,

(15)

= - grip ~y n

or in vector form, ~)Vn/aZ

- VpP]Xk

=

~

N2

kxVnP

(16)

Neutral-surface potential vorticity

195

where [Vn N - VpP] has simply been expressed in terms of the gradient of pressure in the neutral tangent plane, Vnp, by using the hydrostatic equation. These equations confirm that i t is the neutral tangent plane whose slope is dynamically important in causing the horizontal velocity to change in the vertical (thermal wind), and so i t is (approximately) neutral surfaces that should be used as the interfaces in layered ocean models.

The small terms proportional to the slope of an isobaric surface, Vpp, are usually

neglected under the Boussinesq approximation.

I f one uses the slope of potential-density

surfaces in order to infer the thermal wind, one incurs an error in the estimated thermal wind that can be calculated by using equation (8), [VOR - VnN] = Vno [ ~ - I ] / 8 z.

When one

is 1000m away from the reference pressure, IVaR - VnNI can be as large as IVORI at some locations, so that even the sign of vz or uz can be different to that based on the slope of a potential-density surface (see for example, Fig.3 of McDOUGALL1987c).

5.

THE CONTINUITY EQUATION

Although neutral surfaces are formally i l l - d e f i n e d , neutral tangent planes do exist (McDOUGALL and JACKETT, 1988), and the continuity equation can be expressed by considering the flow between two neutral tangent planes separated by a height, h, together with the dianeutral velocities eu and e L across the upper and lower neutral tangent planes.

For a steady-state

ocean, continuity is given by @ hu T I

a hv + ~ I n

(see Figure 3).

+ [eU - e l l

:

~n

[h~n] + [eu - e l l

= 0

,

(17)

n The h e i g h t ,

h, is taken to be v a n i s h i n g l y

small in t h i s paper, so t h a t t h i s

e q u a t i o n becomes Vn.~n[In(h)]

(18)

+ Vn.Vn + e z : 0

In Cartesian coordinates, continuity is simply ux + Vy + wz = 0 and here the components ux and Vy are expressed r e l a t i v e to the neutral tangent plane by using equation (3) of McDOUGALL and JACKETT (1988), (viz. Uxl z = Uxl n - NxUz), to obtain wz + Vn.Vn - [~xUz + ~yVz] = 0

(19)

Using the thermal wind equations (14) and (15), the square bracket here becomes N2[NvPx_ - NxPy]/f, containing only terms that would vanish under the Boussinesq approximation. The geostrophic equations can be expressed as -fv = -gPx and fu = -gPy so that the above non-Boussinesq term can also be expressed as g-IN2Vn.VnN.

In this form, the non-Boussinesq

term is readily seen to be two or three orders of magnitude less than wz, so we are j u s t i f i e d in making the Boussinesq approximation in the rest of the paper.

Combining (18) and (19)

the desired form of the continuity equation for steady flow is found to be,

196

T.J. McDoUOALL

~ ~ " "~x(East)

Two neutral surfaces

It

FIG.3.

Sketch showing two neutral surfaces separted by a height h(x,y). Mixing processes can cause fluid to flow vertically through these neutral surfaces with the velocities e~ and eu through the lower and upper neutral surfaces respectively.

I

I

Wz

: ez + ~n'~n[In(h)]

[

(20)

Note that this form of the continuity equation does not require that the fluid is homogeneous between the pair of neutral surfaces, since the l i m i t h ÷ 0 has been taken. Equation (20) can also be derived by vertically differentiating equation (9) and using the geometrical identity, a[VnN]/Bz = Vn[In(h)].

6.

NEUTRAL-SURFACEPOTENTIAL VORTICITY

The geostrophic equations, -fv = -px/p and fu = -py/p are cross-differentiated to eliminate pressure and the continuity equation, ux + Vy + wz = 0 is used to find the conservation equation of linear v o r t i c i t y in Cartesian coordinates (under the Boussinesq approximation), B T v = wz

(21)

The linear v o r t i c i t y equation can now be expressed in terms of our neutral-surface framework by using the link between the reference frames provided by the continuity equation (20), so that (21) becomes I ~ v = ez + Vn.~n[In(h)]

or

VR.VR[IR(~)] :

ez

I

(22)

This equation implies that in the absence of mixing processes (i.e. for ez = 0), f/h will be conserved following a neutral trajectory. The neutral-surface potential v o r t i c i t y (NSPV) is then defined to be proportional to f/h with a proportionality constant of hiN~; the subscript~ I indicates that these values are evaluated at a reference cast, cast I (where NSPVI = fiN~).

In this way NSPV ~ fN2 L - - T ] hN L

,

satisfying,

~n "~n [In(NSPV)] = ez

(23)

Neutral-surface potential vorticity

Finding an expression for the variation of [hiN~/hN2] ~ ject of the next section.

197

along a neutral trajectory is the sub-

Notice that further use of the continuity equation in the form

(18) transforms (22) to Vn.[fVn] : 0

(24)

This equation applies even in the presence of mixing processes, and i t implies that

the

lateral transport of total potential v o r t i c i t y , integrated v e r t i c a l l y between two neutral tangent planes, is nondivergent (as pointed out by HAYNES and McINTYRE, 1987). This form of the conservation statement of potential v o r t i c i t y does not have the vortex stretching term caused by mixing processes appearing e x p l i c i t l y ; the lateral velocity divergence term ~n'~n"

instead, this term is absorbed into

Note that neither (22) nor (24) is cast in the

form of an advective change of potential v o r t i c i t y following a f l u i d parcel, since a given f l u i d parcel actually flows through our neutral surfaces at a vertical velocity, e.

Rather

than representing a material derivative of potential v o r t i c i t y , these two-dimensional conservation statements describe the variations of NSPV i n

a sur£aoe.

Note that turbulence measurements can be used to predict e and perhaps i t s vertical derivative, ez, but i t is clear from (22) that knowledge of ez alone is insufficient to deduce the meridional velocity, v. For example, consider a gyre that is homogenized in potential v o r t i c i t y (RHINES and YOUNG, 1982), where equation (22) implies that ez must be zero.

Since the meri-

dional velocity, v, is not zero here, this is a region where the difference between wz and ez, namely, ~n'~n [In(h)] (see equation 20), is important. different to ez in the upper I km of the ocean.

Probably wz is generally quite

In the deep ocean, NSPV maps are dominated

by the variation of the Coriolis frequency with latitude, so that there wz~ ez.

7.

THE VARIATION OF hN2 ALONGNEUTRALTRAJECTORIES:TWO DERIVATIONS

In order to map lateral variations of neutral-surface potential v o r t i c i t y , NSPV, one needs to know the vertical spacing, h, between adjacent neutral trajectories.

One could map two

neutral surfaces and simply difference their pressures at each location in order to find h; however, i t is preferable to find NSPV by mapping only one neutral surface and using the available information about the vertical 9radients of properties at each point on this surface. In this section, an expression is developed for h as a function of lateral position, in terms of the lateral variations of properties along a single neutral surface, but f i r s t a very simple and important relationship between the square of the buoyancy frequency, N, and the vertical gradient of potential density is derived g'IN2 = e(S,O,p)Oz - B(S,O,p)S z,

for use below.

N2 is given exactly by

(GILL, 1982), while the vertical gradient of potential

density (referenced to pr ), @pO/@z, is equal to -pe[~(S,O,Pr)Bz - B(S,O,Pr)Sz]. tions imply that N2 =~ [_g BPo] ~(S,O,p) PB ~ B(S,O,pr)

These rela-

(25)

198

T.J. McDouOALL

The factor

p = C[Rp - l ] / [ R p - c] occurs repeatedly in tracer studies (see equation 2) and

in dynamical studies on neutral surfaces (equations 7 and 8). The expression for the lateral variation of hN2 is derived in two separate ways below. The f i r s t method uses Stokes's theorem on the vector mVe - BVS for an integration path in the vertical plane, while the second involves a detailed audit of potential density variations around the same loop. Both derivations are self-contained so the reader may choose between them based on his taste for either a short but more mathematical proof or a more straightforward but lengthier one. Of course, he may choose to feast on both! 7.1

An expression for hlN~/hN2 by using Stokes's theorem s

Let us examine the path~ependence associated with integrating .VO - BVS around a loop in the vertical plane as shown in Figure 4.

The line integral around the closed path £-u-a-b-£

in Figure 4 is

..~

- ~s].

ar:f~

g-IN2az-fl

g-IN2 az

+f£b ['~n8 - 8~ns]'d~ 7 ~

(26) [.~n0 - 8VnS]. a~

As the sections of the closed path from b to £ and from a to u are along neutral trajectories. the second line of this equation is zero.

Using Stokes's theorem, the closed integral can

be expressed as an area integral over the vertical area £-u-a-b-Z of the curl of o~ZO - B~ZS. Here the expression for the curl of b[ aVO - BVS] from McDOUGALL and JACKETT (1988) equation (8) is repeated: by way of reminder, m is normal to the neutral tangent plane and b[x,y,z] is a general function of space. Vx { b[.VB - BVS]} = b8 B[./B @p ]

VnPXVnO+ g-IN2 Vbxm (27) Bp

Pz - ~np{ ~-pOz - B-p

Setting b[x.y,z] = I . Stokes's theorem and equations(26) and (27) give

u g_lN2az -

g-lN2dz

=

i X { V n 9 o Bp

Pz - ~nP{B'p0z

@-pz ~" ~

where the area element aA has a zero v e r t i c a l component so that~nPX2n0.a ~ = 0. l i m i t as the thickness between the two neutral t r a j e c t o r i e s is shrunk to zero, mX~n0, aA = -Vn0.[mxdA] ÷.-h~02_ "a£~ and mx~~.p.aA + (-pz)h~nU.d~, (28) tend to hg'lN 2 and h l g ' l N 1, so that ,. -

:

ngm ~

(-pz)VnO .d~

the v e r t i c a l

, (28) Taking the

integrals

3, 0z - ~-~ @BS} hg{ ~-~ z (_pz)Vn,V.ag.,. ~ ~

in

(29)

Neutral-surface potential vorticity

199

m/

%/

-C

d,

j

dA

hi

_L L.___

T h

neutral trajectories

P cast 1 FIG.4.

A vertical cross-section showing two neutral trajectories and the height, h, between them. The area element, aA , for the area a-b-£-u is directed out of the page, while the two-dimensional line element, a~, is also shown in this figure.

This is a f i r s t order integral equation for h/h I with d~ = (dx,dy,0) being the independent variable(s), and its solution is N2 H-N-2~@[ ~/B] f~ In[hlh -~] N =-Ya ~ P p-T~---(-Pz)V~nO'd~+ gN-2{~

0z

- 8~S ~F~ z } (-Pz)~ N'd~,

(30)

The expression R8[~/8] is the thermobaric parameter of McDOUGALL (1987b). Neglecting the Bp variations of 8 with pressure in comparison with those of ~ in the curly brackets above, and putting k = glpzlBB[s/B]/ap:2.6 x 10-7s-2K-I,(30) simplifies to

Eh } l =exP{-kf: h N2XRO +kfu ,

(31)

Another way of deriving this result is to ask what function b[x,y,z] makes b[~VB - 8vS] pathindependent for integration paths in the vertical plane. By Stokes's theorem, this requirement implies that the horizontal components of Vx{b[~B - B[S]} must be zero, and from (27) a differential equation in two dimensions is obtained for b with the solution (to within a multiplicative constant), b = [h~N~/hN2] as given by equation (31). T h i s analysis has pinpointed the path-dependence of N2 in the v e r t i c a l plane as the reason for neutral-surface potential vorticity not being simply proportional to fN2: a form of path-dependence f i r s t realized by VERONIS (1972). While the vertical integral of bN2 between two neutral trajectories is independent of position, the vertical integral of N2 is not. This type of pathdependence can occur as a result of either the lateral variations of potential temperature along a neutral trajectory, or the slope of a neutral surface, VnN (see equation 31). By way of contrast, path-dependence in the definition of a neutral surface relies on the specific cross-product of two horizontal vectors, ~nPX~nS.

200

T . J . McDOUGALL

In the very simple situation where potential temperature does not vary on neutral surfaces, that is, where Vne = 0 and @[VnO]/@z = O, potential density is constant on neutral surfaces and the function b that makes Vx{b[=VO - BVS]} path-independent can be shown to be

b:

r hi

r!

h

~]

Cs,o,p/

= PO (S,O,pr)

where IJ = C[Rp- 1]/[Rp - c].

(32)

IJ

The vertical component of b[eVO

BVS] is then equal to

-BpO/BZ.

7.2

Finding hlN2/hN2 by keeping track of potential densitg

Variations of potential density are given in terms of increments of s a l i n i t y and potential temperature by dp0 = pO[B(Pr)dS - a(Pr)dO], so that the change in potential density along a neutral trajectory (along which ~(p)dO = B(p)dS) is equal to _[Poe(Pr)[C-1]dO" This integral is performed along a neutral trajectory and can be expressed alternatively as

finpoa(Pr)[C-1] Figure 5(a)

A vertical cross-section through two neutral trajectories is shown and a plot of pOe(pr)[C-1] against potential temperature, O, along these two neutral trajectories in Figure 5(b). Points a and b l i e in the upper and lower neutral tra-

jectories at an i n i t i a l cast, (labelled cast I ) , while points u and £ are on these same neutral trajectories some distance away. To a good approximation, pea(Pr)[C-1] = 2.7 x I0 -5 [p-pr ] kg m-3 K- I ,

(33)

(see McDOUGALL, 1987a), where pressure is measured in decibars. density along the upper neutral trajectory is

fu

POu - P%a =

pO~(pr)[C-1]dO ~ 2.7xi0 -5

fu

a

The change of potential

[p-pr]dO

,

(34)

[P-Pr]dQ

,

(35)

and along the lower neutral trajectory is - Po

=

pe~(Pr)[C-1]

dO = 2.7x10 -5

where p is in db and 0 is in °C. b a Let 6Po1 be the difference in potential density between points b and a at cast I (6P01 = pe-pe)

and l e t 6p0 = P~ - PO u " The difference between (35) and (34) can be interpreted in terms of the areas on Figure 5(b) by (P%£ - POb) - (P8u " POa) = 6Po - 6P01

=

AL - AR + A

(36)

The areas AL and AR are given by

AL = P O l a l ( P r ) [ C l - 1 ] e z l h l

,

(37)

Ic)

S

p

neutral trajectories

( ° ~ e ,-r---__

~

neut---

vertical cast

r

q

p'

6Z

Cd)

p

or

Po ct(p,) [c-ll

(b)

/

~werl

trajec~

Z/

ve~cal c~

/~

b

:}

~<{!

vertical cast

Area A

/ neutral trajectories

vertical cast

1

/ °

: 2::

AreaAR..,

FIG.5. Vertical cross-sections along the direction ~ of integration, in physical space (figures a and c), and in @-p space (figures~b and d). Figures c and d are expanded views of the central parts of figures a and b respectively. The vertical axes in figures b and d may be r~garded as pBo(p~)[$-1], since this is approximately proportional to 2.7 x 10- b [ p - p r ] kg m-~K- l . Note that the difference in pressure between points r and s in figure b is not h but 6p db (= ~z = zs - zr while h = zq - zr).

d

•

"°O'r~,

1

7 ............... ............2___

~

./ip

o surfaces

isobar (p = constant)

cast 1

h'

~ ~ - . ~ P e r

(a)

g

g

Z

202

T. ,J. McDOUGALL

AR : p0a(Pr)[C - I] Bzh

(38)

,

while the vertical differences of potential density, 6pB1 and SpQ can be expressed as (using BpB/Bz = pB[B(Pr)Sz - ~(pr)ez]) Spel = pel~l(Pr)[1 - CllRpl]QZlhI

(39)

SpQ = pB~(pr)[1 - clRp]Bzh

(40)

The limit hI ÷ 0 is assumed and our aim here is to find an expression for the ratio h/h I. Equation (36) can be simplified using (37-40) to obtain C[Rp- I] ape--~

~PB1

c1[Rpl- I] [Rpl- c I]

:

A

(41)

Combining (25) and (41), and taking hl, and hence h, to be vanishingly small, we have (assuming B(p)/B(pr) = I) hN2 - hiN ~ :p~iBA .

(42)

Now we turn attention to evaluating the area A. If 2.7 x 10-5 ap is the vertical difference in the ordinate of figure 5(b), measured at a given value of potential temperature (see the expanded view in figure 5(d)), then the area A is equal to 2.7 x 10-5 -[u ~P de kg m-3. The pressure difference, ~p (or equivalently, the depth difference, ~z), c~n be found from the expanded ocean cross-section shown in Figure 5(c) in terms of the vertical distance, h, between neutral trajectories. The tangent of the angle ~ between the neutral trajectory and the geopotential is equal to the dot product of - VnN with the direction £ of integration around the neutral trajectory. The ratio of [6z - h] to h is equal to the ratio of -VnN.d£to VnS.d~/Bz, and so ~z VnO.d£ : h Vn0. a£ - h0z VnN .a£

(43)

The area A is then given by A= 2.7 x I0 -5 i ~ h Vne .d~- / ~ hez v-n N.d£} kg m-3

(44)

Equations (42) and (44) now give an integral equation for the variation of h/h I with lateral position along a neutral trajectory, hNI2 h ~ __

:

2"7 x I0-5 { f ua h VnB .a£ - f u hBzV n I + Peg-lhiN~. ~ a ~

the solution of which is equation (31) above.

N.d~ } ,

~

(45)

Neutral-surface p o t e n t i a l

8.

vorticity

203

NSPV COMPAREDWITH IPV AND fN 2

The isopycnal potential v o r t i c i t y (IPV), defined by IPV = -fg[BPo/BZ]/PO' is related to fN2 (from equation 25) by I fNz

B(S'B'p) = 6(S,B,pr )

P ~ ~

I

(46)

since the saline contraction coefficient, B, varies l i t t l e

with pressure.

In the special

case where potential temperature does not vary along neutral surfaces ( i . e . v-nO = Q everywhere), PB is also constant along a neutral surface so that potential-density surfaces coincide with neutral surfaces and NSPV is simply proportional to IPV. From (46) i t can be seen that fN2 is not proportional to IPV and NSPV, but varies along the neutral trajectory because of the lateral variations of u.

As an example, i f a neutral trajectory dips down from the

sea surface where p = Pr = 0 to 1500db where 0 = 6°C and Rp = 1.85, U changes from I to 2, implying that fN 2 shows twice the lateral variation as does IPV and NSPV in this situation. In the more general situation where VnO ~ O, equations (23) and (31) may be combined to give the relation between NSPV and fN2; namely, NSPV = b = [ hl

N~- ] = exp { - k

h

fu

N-2VnO.aJL +

a

~

~

kfU

OzN-2Vn N.a~. }

a

~

,

(47)

~

while the ratio of NSPV to IPV (both measuredfollowing the vertical excursions of a neutral trajectory) is .--~-] = ~

exp{ -k

v-nO .d~ *

a

OzN-2V-n e.d~}

(48)

IPV and fN 2 are usually contoured in potential-density surfaces and so there are further differences between a map of NSPV on an (approximately) neutral surface and the usual maps of fN2 and IPV because of the vertical separation between neutral surfaces and potentialdensity surfaces. Taking N~ = 2.7 x 10-6 s-2 (appropriate to a depth of about, 1500m) and a change of ~ along the neutral trajectory of 2°C, l~ads to a value for exp{ -k~N-2VnO .a~ of about 1.22. The 2 J ~ other exponential term, exp { kfa OzN- Vn N .a~ , can have ~n additive effect of about the same magnitude in (47), so that NSPV/fN2 can be as large as 1.222 = 1.5 after having been I at a reference cast where the neutral trajectory was two degrees cooler and perhaps 1000m deeper. The ratio NSPV/IPV has the extra factor of U = C[Rp- l]/[Rp- c] multiplying the same exponent i a l expression, and so i t may vary by an even larger factor. These simple estimates demonstrate that the subtle nonlinearities of the equation of state of seawater have important implications for the calculation of potential v o r t i c i t y in the ocean. More definitive demonstrations of the distinctions between approximately neutral surfaces and potential-density surfaces for dynamical ocean studies w i l l have to await detailed plotting from the large hydrographic data sets that are now available.

204

T . J . McDoUGALL

9.

THE BETA-SPIRALMETHOD

The beta-spiral method of determining absolute velocity vectors in the ocean was pioneered by STOMMEL and SCHOTT (1977) and SCHOTT and STOMMEL (1978).

In these papers the slopes of

potential-density surfaces or steric anomaly surfaces were used, while, as Davis (1978) pointed out, in s i t u

density, p, was taken to be a conservative quantity.

SCHOTTand ZANTOPP

(1980) included vertical mixing in the beta spiral method, but without addressing the problems caused by the compressible nature of sea-water.

In this section, the beta spiral equations

w i l l be developed bearing in mind both lateral and vertical mixing processes and the f u l l y non-linear nature of the equation of state of seawater.

I t should not come as a surprise

that the relevant surfaces whose slopes should be used in the beta-spiral method are neutral tangent planes. SCHOTT and STOMMEL(1978) demonstrated that the s p i r a l l i n g of the lateral velocity with depth is proportional to the "vertical" velocity, and then developed a method for the least-squares solution of reference-level lateral velocities. here.

The same two-stage procedure is followed

First we form -uv z + VUz, l e t t i n g u = U(z)cos(@), v = U(z)sin(~), and use the thermal

wind equations, (14) and (15) together with (9) and the geostrophic equations -fv = -gPx and fu = -gPy, to find N2 N2 -UVz + VUz = -U2~z = f - ~ n "~n N : T - [w-el

(49)

This equation shows that, having included all the nonlinear terms of the equation of state, the essential beauty of the B-spiral remains. The sense of rotation of the lateral velocity with depth, -@z' while not being proportional to the total vertical velocity, w, is proportional to [w-el.

That is, "~z is proportional to the slope of the neutral surface in the

downstream direction, [Vn.VnN]/U.

I f Vn N = O, the lateral velocity vector w i l l not spiral

with depth even though w and e are both non-zero.

Also, i t may come as a surprise that,

for a given lateral velocity vector, and for the observed slope of the neutral tangent plane, the s p i r a l l i n g of the lateral velocity vector with depth is t o t a l l y independent of mixing processes, since they contribute equally to both w and e, implying that there is no signature of these mixing processes in [w-el (see figure 2).

The many processes that contribute to e

(see McDOUGALL 1984 and 1987b), including isotropic small-scale turbulence, double-diffusive convection, cabbeling and thermobaricity, do not affect -U2~z. Rather, -~z indicates whether the sliding motion of the lateral component of the total f l u i d velocity gent plane

can be quite different to of

along a neutral tan-

has an " u p h i l l " or "downhill" component through the term [~n'~n N]/U"

SinceVnN

gCR, the predicted lateral velocity component in the direction

VoR, based on the observed rate of turning with depth of the lateral velocity vector,

-@z" may be quite different i f one performs this type of study in potential-density surfaces (see equation 8 and Figure 6). The relevant equation for the determination of absolute velocities by the beta spiral method comes d i r e c t l y from the linear potential v o r t i c i t y equation expressed in neutral tangent plane coordinates, equation 22,

Neutral-surface potential vorticity

~ ~

P

0

205

= constant,z = ~ [x,y] v .-..Fk---- geopotential neutral tangent plane, z = N [ x , y ] I ~

FIG.6.

0 = constant

The slope of a potential-density surface may have a different sign to that of a neutral surface, as shown here. The "B-spiral" rotation of the lateral velocity vector with height, ~z, depends on Vn.VnN. The figure shows the vertical plane containing Vn, and i t i l l u s t r a t e s a case where V . VnN has the opposite sign to V~.V ~ , which would be obtained from a study carried out with potential-dens~'ty~surfaces. Also the difference between the slopes of these surfaces affects the thermal wind predicted by studies in the two different reference frames.

hx u { ~--} + v { h

h

h

-

=

where the lateral velocity components have been s p l i t into reference level velocities, (Uo,Vo), and geostrophic velocities, ( u ' , v ' ) .

In terms of the vector slope of the neutral tangent

plane, VnN = Nxi +Nyj, (50) can be expressed as

u.-N'xz + V [ N y -

Bzlf]z

+ u'IVxz + v ' [ ~ -

(51)

Bzlf] z + ez = 0

While the function z = ~[x,y] defines a single neutral tangent plane, by considering a series of such tangent planes in the v e r t i c a l , i t is clear that ~x and Ny are well-defined functions of three-dimensional space, and ~z and Nyz in the above equation are to be interpreted as the components of @~nN/Bz. Mixing processes of various kinds enter these equations through the term ez. I t is apparent from (50) and (51) that i t is the slopes of neutral surfaces that are relevant to the B-spiral method. I f other surfaces (e.g. potential-density surfaces, potential isotherms) are used, the term involving the interfacial velocity component (wdz or w~) cannot be interpreted as resulting from mixing processes. I t is well-known that the B-spiral method becomes ill-conditioned i f the potential v o r t i c i t y is constant over some lateral area; this is readily demonstrated for the neutral-surface potential v o r t i c i t y .

With NSPV locally constant on a neutral surface, Vn[f/h] = 0, implying

that hx = 0 and hy/h = B/f. From (50) the coefficients of both Uo and ~ case, explaining the degeneracy. Since the slope of a potential-density surface,

are zero in this

~oR, can be significantly different to the

slope of a neutral tangent plane, VnN, (see equation 8) we must expect that the vertical derivativeS, Nxz and Nyz, that enter the B-spiral regression equations, w i l l also be s i g n i f i c antly different to the corresponding quantities based on potential-density surfaces, R xz and Ryz. Here an estimate is made of the relative sizes of these terms when the B-spiral technique is used with neutral surfaces or with potential-density surfaces. By taking the lateral spatial derivative of (47) and (48), i t can be shown that

206

T . J . McDouOALL

-Vn[In(NSPV)] - ~n [In(fN2)] = -kN-~nO + kOzN-2~n~

'

(52)

and ~n[In(NSPV)] - ~n [In(IPV)] = -kN-2VnO + kOzN-2Vn~V+ Vn[In(lJ)]

(53)

While IPV and fN 2 are usually plotted in a potential-density surface, these equations are concerned with gradients of these properties in neutral surfaces.

Notice from (52) that

the differences between lateral maps of NSPV and fN 2 are independent of any reference pressure of a potential density variable. The terms on the right-hand side of (52) can be estimated by taking typical values of 19nOl ~ 10-6 K m- I , IV,NI ~ 10-4 , N2 ~ 3 x 10-6 s-2 and noting that OzN-2is simply g - I ~ - I [ I -"Ro-I] - I .

The term -kN-2Vn0 is thus of the order I x I0-7m- I

while the second term, kOzN-2VnN is about a third of this magnitude.

These values are to

be compared with the planetary gradient of potential v o r t i c i t y that appears in (50) and (51) in the term B/f, which is about 2.3 x 10-7 m-I in mid-latitudes. Note that in order for the relative variations of NSPV to be proportional to those of fN2 along a neutral trajectory, 9n0 must be parallel to Vn~ and these gradients must satisfy 9nO = ~z~n~ (from 52), which implies that the surfaces 0 = constant must be horizontal; a very special case indeed. When potential temperature does not vary along neutral surfaces, so that potential-density surfaces and neutral surfaces coincide, we expect that the right-hand side of (53) w i l l be zero.

This is indeed the case and can be proven by taking ~n0 = O, @[VnO]/az = 0 and expand-

ing Vn[In(~)] while regarding c as a function of S, 0 and p. Most B-spiral studies to date have used the slopes of potential-density surfaces, and so a linear regression equation like (51) is used with @[V~R]/BZ in place of @[gn~]/@z as the coefficient of the velocity components.

I t is shown below (see equation 56) that the con-

servation equation of IPV in a potential-density surface actually contains an additional term, ~nPX~nO.~ [ I R ~ 1 ] e [ u 1]/[fp ]. Here however we simply compare the coefficients @[VoR]/az and a[gn~]/az by using (8) to find a [9oR]Iaz

- a[VnlV]laz

=

[u - 1]a[Vn01Oz]laz

+ [Vn010z][a.~laz]

(54)

The f i r s t part of the right-hand side of (54) is d i f f i c u l t to estimate, but the angle between potential isotherms and neutral surfaces may vary by 10-4 over a vertical depth range of 1000m, so that @[~nB/ez]/@z may be of order I0-7m- I .

I f one uses a potential density variable

referenced to a pressure lO00db distant, then u ~ 1.3, giving an estimate of the f i r s t term on the right hand side of (54) at 0.3 x 10- 7m-I. The second term in (54) can be expanded by differentiating u = C[Rp- 1]/[Rp- c] with respect to z.

I t also has terms that are pro-

portional to the pressure excursion from Pr' but in addition, i t contains the term, -[u2/c] kN'29n ~ e, which reduces to the more familiar term -kN-2VnB when p = Pr' and so is of order I x I0-7m-I or larger.

ARMI and STOMMEL (1983) avoided this term by continually changing

the reference pressures for their potential density variables.

Neutral-surface potential vorticity

207

The scale analysis of this section suggests that even when one uses a locally referenced potential density, errors of order -kN-2VnB s t i l l

enter the B-spiral regression equation,

because of the lateral variations of potential temperature, and that these errors are almost as large as the other dominant term in the B-spiral regression equation, B/f.

Errors of

similar magnitude occur by using a distantly referenced potential density variable when the reference pressure is about 1000db shallower or deeper.

I t is probably more importanz to

use the correct surfaces for obtaining the variations of their slopes in the B-spiral regression equation than to include the term ez caused by the vertical divergence of the total buoyancy flux across neutral tangent planes. density surfaces for the

ARMI and STOMMEL (1983) have defined their

B-spiral technique to be potential-density surfaces, referenced

to the average pressure of all the data on each surface.

In this way, each of their surfaces

very closely approximated a neutral surface over a lateral extent of about 1000km, and the maximum vertical separations between these centrally referenced potential-density surfaces and the neutral surfaces are estimated to be only 20m.

10. 10.1

CONSERVATION STATEMENTS IN POTENTIAL-DENSITY SURFACES

Nonconservation

o f I P V on p o t e n t i a l - d e n s i t y

surfaces

The continuity equation with respect to potential-density surfaces has a similar form to that with respect to the neutral tangent plane (equation 18), so that Vn'Vo[In(hÙ)] + ~o'~n + wd = 0, where h~ is the height between successive potential-density -Z surfaces.

The link with the Cartesian term wz comes from expressing ux + Vy with respect

to a potential-density surface as Vo.Vn - [RxUz + ayVz], similar to equation (19). In this case, however, the term in square brackets contains not only the non-Boussinesq term, g-IN2Vn. VnN, but also the term [Uz, VZ].[R x- NX, R y - N y ] , which can be evaluated from the thermal wind equations, (14 and 15) and equation (8), obtaining

~o.Vn - Vn.Vn : VnPXVnO.k[I - RpI] o[~f; 13

(55)

The linear v o r t i c i t y equation (21) then becomes

I ~- V = wd + VR .Vo [In(h(:I)] + VnPXVnO.k [I - Rpl] °f-~ Z

~

I

'

(56)

or Vn.Vo [In(IPV)]

d + VnPXVne.k [ I - RpI ] = wz

-

~

,

(57)

or, in the divergence form

Vo.[fV n] = VnPX~ne.k[1- R~1] ~

(58)

208

T . J . McDouoALL

The extra term that appears in these conservation equations for IPV in potential-density surfaces is reminiscent of the term ?X.Vpx?p/p3 that appears in the usual conservation equation of Ertel potential v o r t i c i t y following a f l u i d parcel (PEDLOSKY, 1979, equation 2.5.7). When potential density, PC' is used for the "conservative" variable, X, VX .Vp xVp/p 3 is nonzero. By using the f u l l derivatives, ?Pc =pB[B(pr)?S - e(Pr )re] and Vp = ?into ?- + ~--I m, one flnds that p[B(P)VS + y(p)?p], and ~ . - ~(p)V8 . . . s p l i t t. i n g n a dz x,y~ VpO.Vpxvplp3 :

VnPXVnO.k [ I - R~1]eEu - 1][aPolaZ]Ip2

(59)

The neutral tangent plane framework avoids this term because the tangent plane is normal to [~VO - B? S] = -[~-Vp - YVp], so that the dot product of this vector with VpxVp is zero. The presence of these terms proportional to ?nPX?nO.k implies that IPV is not conserved along potential-density surfaces even when w~ is zero.

The magnitude of this non-conservation

can be evaluated by taking Vnp to be [-gp] x 10-4; the epineutral gradient of potential temperature, ?nO, to be 0.5 x 10-6 K m-I", ~ to be 2 x 10-4 K-I; and [ I - R -1][p _ I ] to be about ~

p

0.5 (consistent with being 1000db away from the reference pressure of the potential density variable); implying that ?nPXVnS.k [ I - Rpl]~[p - 1 ] / [ f ~ is the same size as Bv / f in (56) i f v is 2mm s- I . As this is not a small lateral velocity, i t is a further caution for the use of potential-density surfaces and isopycnal-potential v o r t i c i t y in dynamical studies of the ocean circulation.

The magnitude of VnPX?nOok can also be evaluated from maps of

pressure and potential temperature in a potential-density surface, since VoO = ~V~

(from

equation 2) and [?a p - ?np] = -gp[?aR - ?nN] = -gp?nO[~ - I]/8 z (from 8), so that ~px~O.k

=

~oPXVaO.k/u

(60)

From (56)-(59) i t may appear that i f ?nPX?n8 : O, potential-density surfaces would be satisfactory for dynamical studies of the ocean circulation. However, this is not the case for four reasons. First, unlike the dianeutral velocity, e, the diapycnal velocity is not interpretable solely in terms of mixing processes (see ( I I ) above and (71) below). Second, the thermal wind is not proportional to kXVaP, but instead can be expressed as (from (8) and (16)) aVnlaZ

'21

= ~--

kxVapl[gp]

+

kxVoO[1 - I / u ] / O z

1

(61)

When one is 1000db away from the reference pressure of the potential density variable, the extra term r e s u l t i n g from kX?aO here w i l l often be larger, and in a d i f f e r e n t l a t e r a l direction,

to the kx?o p term.

to determine the v e r t i c a l correct l a t e r a l

Layered models that use the slopes of p o t e n t i a l - d e n s i t y surfaces shear of the l a t e r a l

v e l o c i t y vector,

p o t e n t i a l - d e n s i t y surfaces.

v e l o c i t y vector w i l l then not reproduce the

even though IPV may be conserved along streamlines in the

Third, even when Pr = p and so ~ = I , the conservation equation

f o r NSPV is d i f f e r e n t to that f o r IPV in potential density surfaces (see equation 54 above).

A fourth additional problem with such models is that the appropriate reduced gravity is not based on differences in potential density, gAp~p8 (see section 2 and equation 7).

Neutral-surface potential vorticity

10.2

A geometrical

constraint for the conservation

209

of potential

vorticit9

In order to pinpoint the reason for the non-conservation of IPV in potential-density surfaces, i t is instructive to f i r s t generalize the problem. Consider a series of surfaces, each marked by a constant value of some function X = k [ x , y , z ] . A potential v o r t i c i t y variable is defined as proportional to f/h ~, where hx is the vertical distance between two closely-spaced X surfaces.

This potential v o r t i c i t y is conservative i f f Vn .VkEln(f/h>')]

= W~z

or

Ux.EfVn]

where wx is the vertical velocity through a X surface.

= Bv + fVx.Vn = 0

,

(62)

Following similar reasoning to that

in sections 5 and 6 of this paper, i t is possible to show that not only is NSPV conservative (i.e.

~.[fVn ] = 0), but this is also true of potential v o r t i c i t y variables defined between

level surfaces of pressure, p, and between level surfaces of in s i t u density, p.

That is,

~n'~n = ~p'~n = ~p'~n

(63)

In order for the potential v o r t i c i t y variable, f/h x

to be conservative, one requires that

UX'~n is also equal to ~p'~n" The differential operators, UX and Up can be expressed as U - m~B/Bz and U - mp @/@z respectively (McDOUGALL and JACKETT, 1988), where ~ and mP are normal vectors to the

X and p surfaces, having unit vertical components. Using equation

(16) for the thermal wind, one finds that ~

N2 fgp ImP - mX].kXUnp

(64)

The lateral gradients of

p and p in the neutral

_

Ux.V~n - Up.V~n = [mp - mx] .BVn/BZ Now consider the line in space, ~px~p.

tangent plane are proportional (p-lVnp = YVnp, see McDOUGALL, 1987a), so using V = Vn + mB/Bz, i t can be shown that I p~XUp : gN2 mx~p :

N2 kx~p - ~-N2 ~,xV_np + -g-~

,

(65)

and so the vertical component of UpxUp is a negligible Boussinesq term, while the horizontal components of ~px~p are fp2a~n/@ z, that is, they are in the direction of the thermal wind vector.

Combining (64) and (65), and noting that ImP - mL] is a horizontal vector while

~n~XVnp is vertical so that their dot product is zero, one finds ~x.Vn - Vp.Vn = [mp - mX].UPxUPl[fp2] : - m;k .UPxUPl[fp2]

,

(66)

since mP is parallel to Up and so does not contribute to the scalar t r i p l e product. Now the requirement that ~'~n - Zp'~n = 0 for the potential v o r t i c i t y variable f l h x to be conservative can be seen to be equivalent to requiring the normal to the k surface to be perpendicular to the line UP~P in three-dimensional space. T h a t is, the k surface must include the line ~x~p. As pointed out by PEDLOSKY (1979) and GILL (1982), any arbitrary

210

T.J. McDOUGALL

function of p and p, X[p,p], satisfies this c r i t e r i o n .

But notice that the neutral tangent

plane also includes the line VpxVp, since the normal to this plane is ~VB - BVS] which is i d e n t i c a l l y equal to [YVp - ~IVp].

Figure 7 shows the four planes of constant p, p, B, S,

and the neutral tangent plane, intersecting along the two lines, VpxVp and VBxVS. A potentialdensity surface includes the line ~exVS since along this line B and S and hence PB are constant.

One then can see from the figure why IPV is not conserved in potential-density sur-

faces since potential-density surfaces do not include the line VpxVp.

The neutral tangent

plane is the only plane that includes both VexVS (as i t must for the purpose of the mixing of scalars) and also ~px~p (as required for the existence of a conservative potential v o r t i c i t y variable).

There are s t i l l

annoying and unavoidable errors associated with using neutral

surfaces over extended regions of the ocean in that the path-dependence in the definition of a neutral surface implies that any real surface that one finds w i l l not be exactly "neutral". This is considered in section 11 of this paper. neutral surface w i l l

I t suffices to say here that an approximately

approximate the neutral tangent plane in Figure 7 much more closely

than a potential-density surface, t y p i c a l l y by two orders of magnitude, so that i t is f e l t that these complications w i l l be unimportant for the calculation of NSPV.

neutral t a n g e n t p l a n e

FIG.7.

Perspective view of the lines VBxVS, and ~px~p, the lines of intersection of the surfaces of constant e and S, and of constant p and p. Every potential density surface ( i . e . for any reference pressure, Pr) and the neutral tangent plane contain the line VBxVS whereas the potential-density surfaces do not include ~xVp. The neu~tra] tangent plane is the only plane that includes both of these lines.

Neutral-surface potential vorticity

10.3

211

conservation equations for scalars in potential-density surfaces

For completeness, some further deficiencies of conducting studies of mixing processes (e.g. inverse models) in potential-density surfaces are derived in this section.

In the conserva-

tion equations of scalars with respect to neutral tangent planes (McDOUGALL, Ig87b), the dianeutral velocity, e, is not a separate process in the ocean, but is a direct result of epineutral and dianeutral mixing processes. Taking into account the epineutral variations of the height between adjacent neutral tangent planes, the water-mass transformation equation for potential temperature can be expressed in the convenient form (which does not involve

e)

etln + [VR-~Vn(hK)].VnB = KV~B +

D~N 2 e Z~de-3~d2S ~- +

a~ a~ e .Vnp } . K~2 e z {~-~VnB.VRB + ~-~Vn

(67)

while the dianeutral velocity is given by N2 [e - Dz] ~-

a~ a~ = D[~Bzz- BSzz] - K{ ~VnO.Vn 0 + ~VnO.Vn p }

(68)

Here D is the dianeutral scalar d i f f u s i v i t y , K is the epineutral scalar d i f f u s i v i t y and the term a~/aB is the cause of the cabbeling process, while a ~ / a p

causes

thermobaricity.

Note a~/ao here is actually a shorthand notation for [a~/ao + 2[~/B]a~/BS - ~2/82]BB/BS] while a~/ap is shorthand for [a~/ap - [~/8]aB/ap]. has been ignored.

Double-diffusive convection of a l l types

I f the material derivatives that appear in the original conservation equa-

tions of O and S are written with respect to a potential-density surface, one can manipulate these equations to obtain etlo+[~n- ~ ~n(hK)].~oe

= ~KV~B +~

~D9-N2e3e'~-z deL +~K~2 ez { ~-V+ ne'~ne -

-~-~V ~ .V } _np

(69)

and [wd _ Dz] [ _ ! @_~] : - ~ Vn(hK).Vnpe/pe - K~[c - I]V~8 + D[~ezz - BSzz] PB a~ ao - K~ {~-~VnB.~n e + -~Vne.Vn p } B where ~ and B are shorthand notations for a(pr ) and B(pr).

(70)

,

The right-hand side of (69) is

simply u times the right-hand side of (67), so that, for example, an inverse study based on potential-density surfaces would predict a vertical d i f f u s i v i t y that was larger than D by the factor U. of e on

Similarly, cabbeling, thermobaricity and epineutral mixing a l l cause changes

potential-density surfaces a factor of u faster than on a neutral tangent plane.

The diapycnal

velocity, wd, has contributions not only from the vertical d i f f u s i v i t y , D,

and from thermobaricity and cabbeling, but also from lateral mixing along the neutral tangent plane (the f i r s t two terms on the right-hand side of (70)). contribute to the dianeutral velocity, e, in equation (68). wd can be expressed in terms of mixing processes as

These two contributions do not The difference between e and

212

T . J . McDoUOALL

Ee

wd]Bz/[U -

-

I]

=

~Vn(hK) .VnB

KV~B -~ 3 d2S + ~ + UN2eZBdo--;

(71)

B~V # . V n 0 + ~-~nO.Vn 8a + K~2 Oz{ ~ p }

;

which is also equal to the epineutral advective change of potential temperature, ~n'~n O' (see (11)) i f the ocean is in a steady state, i . e . i f Otl n = O. We may treat potential density as just another passive tracer in the ocean and find the conservation equation for potential density in the neutral tangent plane to be ^

^

BPB at I n+ [-Vn - -~Vn(hK)].VnPe= pea[c - I]KV~e + PBe[c - I]D~N203 z B d2S (72) + pe@[c - 1]K ~N2 e z

B~

B~ "~n e + B-p~ no "~np }

{aEE~

'

that is, mixing processes change potential density on a neutral tangent plane at a rate pe~[c - I ] faster than they change the potential temperature. All of these undesirable mixing effects of potential density occur because when Vn8 is non-zero, potential-density surfaces are inclined to the neutral tangent planes along which the epineutral advection and lateral (epineutral) mixing occurs; including the epineutral advection and mixing of PB" 11.

APPROXIMATELYNEUTRAL SURFACES AND LAYERED MODELS

So far in this paper, neutral tangent planes have been considered rather than approximately neutral surfaces.

Since the path-dependence associated with the definition of a neutral

surface increases in proportion

to the area over which one considers such a surface, the

ambiguity of an approximately neutral surface is proportional to the square of the lateral dimension of the surface under consideration (McDOUGALL and JACKETT, 1988).

As this lateral

dimension is shrunk to zero, the neutral tangent plane is attained in which the conservation equations of this paper apply exactly.

For local studies such as B-spiral analyses that

involve only f i r s t order lateral derivatives, the neutral tangent plane is appropriate; however, layered thermocline models such as that of LUYTEN interfaces to be well defined over whole ocean basins.

et a t .

(1983) require the models'

The best interfaces in this applic-

ation are the approximately neutral surfaces developed by McDOUGALL and JACKETT (1988). an approximately neutral surface,

In

aVae is not quite equal to BVaS, but ~ : ~VaB - BVaS and

VaA - VnN = gN-2~, where z = A[x,y] is the height of the approximately neutral surface.

The

thermal wind equation then becomes BVn/@Z = ~

N2

kxVaP + ~ kx~

(73)

The difference between the slopes of an approximately neutral surface and the neutral tangent plane, gN-2~, was found by McDOUGALL and JACKETT (1988) to be about two orders of magnitude less than VnN, so that the kx~ term in (73) is only about I% of the f i r s t term. The vertical shear based on the slope of approximately neutral surfaces is then expected to be a good approximation to the true thermal wind.

Neutral-surface potential vorticity

213

The lateral divergence of ~n in an approximately neutral surface can be related to that in the neutral tangent plane by 9a.Vn - Vn.Vn :

(74)

VnPXe.kl[fp]

The same term appears in the conservation equation for linear vorticity with respect to approximately neutral surfaces, Va.[fVn] = VnPXE.k/p, or ~V

:

W za

+ ~n " ~a [In(ha)] +vnpxc'k/[fp] ~ ~ (75)

= e z + Vn.[a(VaA - ~ ~

where wa i s height

the

~

vertical

between a d j a c e n t

N2

velocity surfaces.

E)/az] ~

through Using

,

an a p p r o x i m a t e l y Vn p ~

[-gp]xlO -4,

neutral

surface

and gN-2~

and h a i s

~ 3x10 - 6 ,

the

the e x t r a

term, VnPX~.k/[fp ], resulting from path-dependence is about 3x10-6N2, so that the change to v caused by this term is about O.Imm s-I i f N2 is I0-5s -2, considerably less than the 2mm s-I correction to v that was estimated above when potential-density surfaces were used. In practice in layered models, the thermal wind equation is based on the slope of the interface, and so the small term, gkx~/f, is avoided in (73). In this case, the extra term that appears in (75) is ~n'~ (rather than ~nPX~.k/[fp]) and this is a negligible term of nonBoussinesq size (in fact i t is -1% of the normal non-Boussinesq term, -g-IN2Vn.~n~ ). Also, since gN-21~I is only of order I% of ~aA, the vertical derivative of ~ is not expected to make a significant contribution to the right-hand side of the second line of (75). Since mathematically well-defined surfaces are required in layered ocean models of the type described recently by LUYTEN and STOMMEL (1986), i t is appropriate here to be a l i t t l e more specific about the use of approximately neutral surfaces in layered models. Combining the continuity equation in the form ~.[haVn] + [wau - wa~= 0 (from (17)), and the linear vort i c i t y equation in the form Bv/f + ~a'~n = ~npx~'~/[fp] (from (24) and (74)~ the layered form of the conservation of linear vorticity with respect to approximately neutral surfaces is ha ~ v : Vn.Vaha + [wau - wa~] + haVnPX~.k/[fP] Following the results of the previous paragraph, the path-dependent terms are

(76)

hereafter

ignored and so (76) becomes simply hBv/f = Vn. Vnh + [e u - e~], which can be recognised as a vertical integral of the ~SPV conservation equation (22) (taking Vn to be depth-independent) since [ J~,

Vn[In(h)] dz = [ ~

JR

@[Vn~]laz dz = [VRN]U : ~h. -

-

~,

Consider a two-and-a-half layer ocean (as in LUYTEN and STOMMEL, 1986), with the layers numbered sequentially from the surface, the third layer stationary, h being the depth of the top layer and D being the depth of the second interface. The thermal wind equation at the two interfaces gives

214

T . J . McDoUOALL

u2 =-~-uI

Oy

= - [g~Dy + g~hy]/f

(77)

;

v2= ~-- Dx

;

vI : [g~Dx + g~hx]/f

,

(78)

where the reduced gravity, g Ii ' across the interface below the i t h layer is given by g~ : pigApBi/pB :

g[BASi - aAei]

and is a function of x and y.

,

(79)

Adopting the notation of LUYTEN and STOMMEL (1986) for the

vertical velocity across the sea surface (We) and across interface I (Ws), equations (76)(78) can be combined to give

B h[g'2D~ 'h ^ + gl x]

g2

T-[hyDx - hxDy] - f-2

g2 B [D-h]g'D - ~--[hyDx - hxDy] - ~ 2 x

=

=

-

ws -

ws

we

,

(80)

,

(81)

equations whose sum is B [g'hhv + g'DDv] f'2 1 ^ 2 ^

= we

(82)

These equations are identical to those in LUYTEN and STOMMEL(1986), except that the reduced I gravities, g~ and g2' are now functions of x and y.

the conservation statements w i l l reduce the i n i t i a l

I hope the straightforward nature of shock for researchers coming to grips

with using approximately neutral surfaces in layered models. Having taken into account the f u l l y non-linear nature of the equation of state, the dynamical equations take familiar forms, but the depth of an approximately neutral surface w i l l , in general, be s i g n i f i c a n t l y different to the corresponding depth of a potential-density surface. In order to find a tractable procedure for including the lateral variations of the reduced gravity in layered models, consider the vertical integral of N2 from interface i+I to i - I , as shown in Figure 8. i

i-i

This integral can be written as

!ri-1

i

i-i

-,Z/i+lg-lN2dz: 2ji+l[( ez-BSz]d, : ,fi+l(,de! F i-I

i

i

i-I

fi+lBdS

(83)

Fi-1

To a good approximation, 2 J i + I e dO - ~iAO and -2J i+l BdS,~ BIAS, wherea i and Bi are the values of ~ and B evaluated at the central interface, and AB and AS are the steps of B and S at this interface. (83).

The reduced gravity, gIAPl/p, across the i t h interface is then given by

Since this reduced gravity is defined in terms of a vertical integral of N2, i t is

also equal to the reduced gravity based on the "local density", p~, defined by VERONIS (1972) as

Neutral-surface potential vorticity

215

i-I

l i

i+l

N;

f n ( p 5 = - g ' l . j ' N 2 dz Ap P

FIG.8.

Sketches of N2 and i t s vertical integral plotted as functions of height, z, through a region containing three approximately neutral surfaces, labelled i - I , i and i+I.

and sketched in Figure 8 as a function of z. glAPl/P = gA(In[p~])

From this figure, i t is clear that

I r i-I .2 d z : ½ (hN2)i + ½(hN2)i+1 ,

= ~Ji+1

(85)

where hi and N~ are the thickness and the average value of N2 of the f l u i d above the ith 1

interface. Putting bi = (hiN~)i/(hN2)i (see (32)), where the subscripts I refer, as previously, to a reference cast, (85) can be expressed as gAp~/p~ ~

T i

+

~

,

(86)

which is a weighted average of the values of b of the two layers straddling the interface in question.

I f , as is common, (hiN~) is taken to be equal for both layers at the reference

cast, then in (86) the average value of the reciprocals of bi and bi+ I can be approximated by b-1 where b is evaluated at the height of the interface, so that

This suggests the tractable procedure of f i r s t mapping a series of approximately neutral surfaces from historical oceanographic data;

each surface chosen to roughly coincide with

an interface in one's model. The reciprocal of b[x,y] that is found for each surface from (47) then multiplies the constant value of 0.5[(hiN~) i _ + (hiN~)i+1] to obtain the appropriate reduced gravity for each interface as a function of x and y.

216

T.J. M c D O U G A L L

12.

CONCLUSIONS

A f i r s t attempt is made in this paper to determine the implications of the neutral-surface framework for dynamical studies of the ocean circulation by deriving the exact dynamical conservation equations in neutral tangent planes under the hydrostatic, geostrophic, Boussinesq approximations.

and

This follows on from the work in McDOUGALL (1987a) where the

gradients of scalars in the neutral tangent plane were related to the corresponding spatial gradients in potential-density surfaces, and the work in McDOUGALL and JACKETT (1988), which addressed the untidy issue of the helical nature of neutral trajectories. eleven paragraphs summarize the salient points of this paper.

The following

Each of the eleven points

addresses separate deficiencies of using potential density surfaces in diagnostic or inverse models of the ocean circulation. I.

The reduced gravity that enters the thermal wind equation at an interface in a

layered ocean model is larger than g Apo/pe (where Pe is potential density) by the factor =C[Rp - 1]/[Rp - c], where Rp is the vertical s t a b i l i t y ratio (Rp =eez/BSz) and c = [~ (p)/B(p)]/[~(pr)/B(pr)]. This result applies to whatever type of interface is chosen to separate the mixed layers of a layered model, and so is independent of the neutral-surface notions that dominate the rest of this paper. 2. An expression is found (equation 8) for the difference between the vector slopes of a neutral surface and a potential-density surface, [VOR -VnN], where R[x,y] and N[x,y] are the heights of a potential-density surface and a neutral surface respectively.

This

expression leads to an estimate of the a r t i f i c i a l part of the diapycnal velocity that is not associated with vertical mixing and advection processes. Rather i t is due to resolving lateral motion, which r e a l l y occurs along a neutral surface (epineutral motion), into isopycnal and diapycnal components. 3.

An exact derivation of the thermal wind equation shows that i t is the epineutral

gradient of pressure in the neutral tangent plane that causes the vertical shear of lateral velocities.

To within the Boussinesq approximation, this is equivalent to saying that the

thermal wind is proportional to the slope of the neutral tangent plane (see equations (14)(16)).

The different slopes of potential-density surfaces and neutral tangent planes then

directly influence the magnitude and direction of the predicted thermal wind in layered ocean models. When the difference between the in situ pressure and the reference pressure is 1000db, IVOR - VnNI can be as large as I VORI, SO that even the sign of vz or uz can be different to that based on the slope of a potential-density surface. 4. Taking into account all the non-linear terms of the equation of state of sea-water, a conservation statement is developed for a new potential v o r t i c i t y variable; namely, neutralsurface potential v o r t i c i t y , (NSPV), defined to be proportional to the Coriolis frequency divided by the vertical spacing between adjacent neutral surfaces. previous forms of potential v o r t i c i t y for two reasons.

NSPV is different to

F i r s t l y the surfaces on which i t

is evaluated are different to potential-density surfaces so that they diverge in the vertical

Neutrabsurface potential vorticity

with respect to potential-density surfaces.

217

Secondly, the potential v o r t i c i t y variable i t s e l f ,

NSPV, is different to both fN 2 and IPV. 5.

Isopycnal potential vorticity,(IPV) is conveniently related to fN 2 through the factor u,

while two different derivations provide expressions relating NSPV to both fN L and IPV (equations (47) and (48)).

This shows that fN 2 does not mirror NSPV due to ( i ) the variation

of the compressibility of seawater with potential temperature acting through a term proportional to ~nB,

and ( i i )

the dependence of the thermal expansion coefficient on pressure,

multiplying another term proportional to the slope of a neutral surface, VnN. these two contributions to NSPV/fN2 depends on any reference pressure, Pr"

Neither of

NSPVis not simply

proportional to fN 2, because N2 displays path-dependence when integrated around a loop in the vertical plane. 6.

When taking into account the non-linear nature of the equation of state in the

B-spiral equations, one finds that i t

is the component of the vertical velocity due to the

sliding motion i n the neutral tangent plane that contributes to the s p i r a l l i n g of the lateral velocity vector with depth; a result that is independent of the mixing a c t i v i t y and of the consequent dianeutral advection.

This confirms that neutral surfaces are the natural co-

ordinate system for dynamical studies of the ocean circulation.

The lateral velocity vector

w i l l not spiral with depth i f the neutral tangent plane is f l a t (Vn N = O) even though the vertical velocity, w, due to upwelling, e, may be non-zero. 7.

The implications of using potential-density surfaces instead of neutral surfaces

in the B-spiral regression equation are quantified and the differences are shown to be s i g n i f icant.

Even for

a centrally-referenced potential density surface, the NSPV conservation

equation is different to that for IPV in the potential density surface.

This difference

is of order -kN-2VnB which w i l l often be a considerable fraction of B / f (see the discussion following equation (54)). 8.

I t is shown that the variation of a/B with pressure prevents IPV from being con-

served along streamlines in potential-density surfaces even in the absence of diapycnal advection and f r i c t i o n a l effects.

In practice, this effect may not be as serious as the use of

potential-density surfaces to infer the thermal wind (see equation (61)). 9.

An additional problem in using potential-density surfaces in layered models (of

both inverse and prognostic kinds) is that potential density is mixed and advected not only by vertical processes, but also by epineutral ones.

In section 10.3 the conservation equa-

tions for tracers are derived in potential-density surfaces,

showing the extra terms that

arise from the epineutral mixing and advection that occur along a direction that is inclined to the potential-density surface. 10.

The path-dependent nature of neutral surfaces (as opposed to the vertical

path-

dependence associated with integrating N2 around a vertical loop) complicates things a l i t t l e , JIM{) 20: 3-1~

218

T . J . McDOUGALL

but not excessively so.

The exact dynamical conservation statements take neat and familiar

forms in the neutral tangent plane framework and the i l l - d e f i n e d nature of neutral

surfaces

arises only when one has to form a surface or an interface over a l a t e r a l l y extensive region. In this case the surface that is found w i l l not quite have the "neutral property" that f l u i d parcels can be moved small distances in the surface without experiencing buoyant restoring forces.

In practice this discrepancy,

while annoying, is small; the differences between

approximately neutral surfaces and potential-density surfaces is the much larger issue. 11.

It

is shown that in order to have a conservative potential v o r t i c i t y variable,

the surfaces one considers must include the line Vp~p, the horizontal components of which are

fp2BVn/BZ.

Potential-density surfaces include the line VexVS but not the line ~P~O, explaining why IPV is not conserved in potential-density surfaces. The neutral tangent plane includes both of these lines and so possesses a conservative potential v o r t i c i t y variable, NSPV, while at the same time having the "neutral property" that f l u i d parcels can be moved e f f o r t l e s s l y small distances in the neutral tangent plane without experiencing buoyant restoring forces.

13.

ACKNOWLEDGEMENTS

I t is a pleasure to acknowledge the many helpful suggestions made by Dr David Jackett during the gestation of these ideas.

I have also benefitted from several illuminating conversations

with Prof J. Willebrand. Graham Wells is thanked for preparing the figures.

14.

LIST OF SYMBOLS

height of an approximately neutral surface, z = A[x,y]. A " b e s t - f i t neutral surface" is one example of an approximately neutral surface. an arbitrary scalar function of space, b = b[x,y,z] ratio of ~/B evaluated at the i n s i t u

pressure, p, to that at the reference pres-

sure, Pr' c = ~(p)/~(pr ) an area element directed normal to the area concerned

aA d~

horizontal line element, a~= dx~ + dy~ + 0k

dr

line element in three-dimensional space, a~= dx~ + dy~ + dzk

e

vertical f l u i d velocity through a neutral surface, (: dianeutral velocity)

f g

Coriolis frequency

h

vertical distance between two closely-spaced neutral surfaces

gravitational acceleration ( : 9.8m s-2) vertical distance between two closely-spaced potential-density surfaces

!,!,k

unit vectors in the Cartesian reference frame

IPV

isopycnal potential v o r t i c i t y , IPV = -fg[BpB/@z]/p 0~ fN2/u

I

height of an interface of a layered model, z = z [ x , y ]

Neutral-surface potential vorticity

219

= glpzl times the thermobaric parameter of McDOUGALL(1987b), i.e. m

glpzlB~ / B ] I B P

~ 2.6 x 10 -7 s-2 K-1 normal vector to a neutral tangent~ ~ plane, m = -VnN + k = gN-2[aV8 - B~S]. The magnitude of m is [I + ~C + N=~]0"5 k =

ma

normal vector to the surface~z =Jm[x,y].

NSPV

neutral-surface potential vorticity, NSPV ~ [hiN~]f/h,~where hI and N~ are evaluated on the neutral surface at a reference cast, cast I. square of the buoyancy frequency, N2 = g[a%z - BSz] = B ~ ~ [- ~ @PBheight of a neutral tangent plane, z=N[x,y] PB @z j

N2 N

P

Pr P R

ma = -Vam + k

pressure reference pressure for the evaluation of potential temperature and potential density. height of an isobaric surface, z = P[x,y] height of a potential-density surface, z = R[x,y] in situ

R

P S

stability ratio, R = aBz/BSz salinity

~n

lateral velocity vector, Vn = u~ + v~ + 0k magnitude of the lateral velocity, U = IVnl = [u 2 + v2] 0"5

U

eastward velocity component northward velocity component vertical velocity (past geopotentials) vertical velocity through a potential-density surface (= diapycnal velocity), d

U V W

wd W

x,y,z

w : e - Vn.Vne[p - I ] I 0 z v e r t i c a l v e l o c i t y through a p o t e n t i a l isotherm (~ diathermal v e l o c i t y ) , e w = e + Vn.Vne/Oz Cartesian coordinates to the east, north and upwards r e s p e c t i v e l y thermal expansion c o e f f i c i e n t ,

a = -

BP/Be]S, p

P B = p~-[BP/aS]e,p = pZ[BO/aS]T,p + a[Be/BS]T,p

B

saline contraction coefficient,

B

meridional gradient of the Coriolis frequency, g = df/dy

Y ~z

compressibility of seawater, Y = P]{BP/@P]s,e : pZ[BP/BP]s,T + a[@B/BP]s,T vertical pitch of a neutral helix; that is, the vertical distance between successive loops along a neutral trajectory vertical distance between a neutral trajectory and a potential-density surface an error vector of an approximately neutral surface, ~_ = ~V~ae - BY~aS = g-lN2[~aA-vn ~] the factor C[Rp - 1]/[Rp - c] in s i t u density, p = p(S,T,p) or p = p(S,B,p), so that--~= BdS - ado +Ydp potential density, P9 = P(S'0'Pr)' so t h a t ~dPB = B(Pr)dS - a(Pr)dB

AZ £ P PB 0

cPz V ~

_Vn

potential temperature, e = 8[S,T,p,pr] rate of spiralling of the lateral velocity vector with height three dimensional spatial derivative operator, V = ~x ly,z~ + ~ y I ~+-~z I ~ • ~z m (see McDOUGALL and JACKETT, 1989) x,z x,y Also, V : ~n + Ix,y~ lateral

gradient operator f o r property changes in a neutral tangent plane,

~Vn = B~-~Ini ~-. ~ + oJ~B-c'I+j~n

0k,~ where o^~B-~n-I indicates that the lateral differences, ~ ,

of a

220

T . J . McDoUGALL

property, @, are evaluated on a neutral trajectory and then divided by the horizont a l distance between, 6x, and the l i m i t , ~x +

~n = ~ - ~ I _VnN

0 is taken.

Note that

x,y

vector slope of the neutral tangent plane to the geopotential plane,

VRN = NX~ + Ny~ V NO

lateral gradient operator for property changes in a potential density surface

Va

lateral gradient operator for property changes in an approximately neutral surface,

vO

~e

which may or may not be a " b e s t - f i t " surface ~a =@~a I ~ + ~al a ~" + Ok.

ma Note that ~a = ~ - ~-~Ix,y~

15.

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