Copyright © 2001 Elsevier Science B.V. All rights reserved.

Progress in Oceanography
Volume 54, 2002, Pages 493-501.

A state variable for characterizing water masses and their diffusive stability: spiciness

P. Flament

University of Hawaii at Manoa, Honolulu, HI 96822, USA.

Abstract

A state variable small pi, Greek (straight theta, small theta, Greek ,s) most sensitive to isopycnal thermohaline variations and least correlated with the density field is proposed. small pi, Greek is constructed so that its diapycnal gradient dsmall rho, Greek small pi, Greek is related to the density gradient ratio Rsmall rho, Greek :dsmall rho, Greek small pi, Greek approximate (1+Rsmall rho, Greek )/(1-Rsmall rho, Greek ). It is simultaneously a tracer conserved by isentropic motions, and an indicator of diffusive stability. It is useful for the combined description of interleaving and double-diffusive processes at the boundary between water masses.

Author Keywords: Equation of state; Double diffusion; Tracer


CONTENTS

1. Introduction
2. Definition
2.1. Spiciness as a state variable
2.2. Diffusive stability
2.3. Constructing spiciness
3. Example
4. Discussion
5. Conclusions
Acknowledgements
References


1. Introduction

The distributions of potential temperature straight theta, small theta, Greek and salinity s on surfaces of constant potential density small rho, Greek (straight theta, small theta, Greek ,s,p=pref) are not independent: displaying both is redundant. Vertical profiles of either temperature or salinity also contains some information on the density stratification. It is therefore natural to introduce a state variable defined to be most sensitive to isopycnal thermohaline variations, and least correlated with the density field. The concept of such a variable is not new; it has been used, among many others, by Stommel (1962) to discuss the sharpness of the straight theta, small theta, Greek –s curve in deep water, by Munk (1981) to separate internal waves from intrusive fine-structure, by Olbers, Wenzel, and Willebrand (1985) as a tracer in an inverse model of the Atlantic circulation.

Flament (1986) defined and constructed such a state variable small pi, Greek (straight theta, small theta, Greek ,s) in a way useful to characterize water masses, and also to indicate double-diffusive stability. Following Munk (1981), small pi, Greek was called `spiciness', being largest for hot (spicy) and salty water. It has been widely used to study the California Current System ( Niiler; Lynn; Simpson; Huyer; Barth and Steger). A number of papers in the present volume use this definition of spiciness, which this note makes readily accessible.

2. Definition

2.1. Spiciness as a state variable

A state variable is associated with a family of isopleths in the straight theta, small theta, Greek –s plane, intersecting the family of isopycnals. A water type can thus be defined interchangeably by its (straight theta, small theta, Greek ,s) coordinates, or by its (straight theta, small theta, Greek ,small pi, Greek ) coordinates. The definition of spiciness should not depend on a specific choice of scales on the straight theta, small theta, Greek and s axes, and must, therefore, be invariant for all geometric transforms allowed in the straight theta, small theta, Greek –s plane. These consist of changes of scales along the axes; rotations are not allowed because straight theta, small theta, Greek and s have different physical dimensions. A scale-invariant constraint is to require that, at any point, the slopes of the isopycnals and of the spiciness isopleths are equal and of opposite sign (Fig. 1):


Image (1)

where partial differential straight theta, small theta, Greek small pi, Greek denotes the partial derivative partial differential small pi, Greek /partial differential straight theta, small theta, Greek of small pi, Greek with respect to straight theta, small theta, Greek . This first order partial differential equation has a unique solution given an initial condition along a line distinct from its characteristics, themselves solutions of the associated ordinary differential equation:

partial differential straight theta, small theta, Greek small rho, Greek dstraight theta, small theta, Greek =partial differential ssmall rho, Greek ds (2)

which defines a unique family of spiciness isopleths, regardless of the scales chosen for the axes.


Enlarge Image (6K)

Fig. 1. A point of the straight theta, small theta, Greek –s plane with the scales chosen so that the isopycnal is at with the axes. The spiciness isopleth at that point is constructed so that ; it is orthogonal to the isopycnal with this choice of scales. A vertical profile unstable to salt fingering is drawn, with curly or open small phi, Greek =-71.6°(Rsmall rho, Greek =2); the arrow indicates the downward direction. The sectors corresponding to profiles diffusely stable, unstable to salt fingering and layering, and gravitationally unstable are also shown. Positive angles are counter-clockwise.

Geometrically, at any point the isopycnals and the spiciness isopleths are symmetric with respect to the straight theta, small theta, Greek –s directions. When the scales are chosen so that the isopycnals are inclined at 45° with the axes at a point, the spiciness isopleths are orthogonal to the isopycnals at that point. Note that (2) is not a total differential, since in general the condition of integrability


partial differential straight theta, small theta, Greek s2small rho, Greek =0 (3)

is not satisfied, i.e. the isopycnals in the straight theta, small theta, Greek –s plane are curved.

The family of spiciness isopleths defined, it remains to label them: to assign a useful value of spiciness to each. Many choices are possible, for example one could select an initial condition small pi, Greek 0(small lambda, Greek ) along a linestraight theta, small theta, Greek (small lambda, Greek ), s(small lambda, Greek ), such as a straight theta, small theta, Greek –s curve, or a particular isopycnal. Instead of an exact initial condition along a particular line, an approximate `initial' condition will be enforced in a least square sense over the domain of the straight theta, small theta, Greek –s plane containing most oceanic water masses. This condition will be selected to relate the diapycnal gradient of spiciness dsmall rho, Greek small pi, Greek to the diffusive stability of the water column.

2.2. Diffusive stability

The parameter controlling diffusive stability is the density gradient ratio (Turner, 1973):


Image (4)

where p is the hydrostatic pressure. Equation (4) provides a simple physical interpretation of (2): the spiciness isopleths are simply the lines along which the density ratio Rsmall rho, Greek =-1 (the adiabatic temperature gradient capital Gamma, Greek has been neglected in (4); this is justified in the upper ocean and the main thermocline, where most water mass interleaving occurs).

The vertical gradients of small rho, Greek and small pi, Greek are:


dpsmall rho, Greek =partial differential straight theta, small theta, Greek small rho, Greek dpstraight theta, small theta, Greek +partial differential ssmall rho, Greek dps=partial differential ssmall rho, Greek dps(1-Rsmall rho, Greek ) (5a)


dpsmall pi, Greek =partial differential straight theta, small theta, Greek small pi, Greek dpstraight theta, small theta, Greek +partial differential ssmall pi, Greek dps=partial differential ssmall pi, Greek dps(1+Re) (5b)

so that

Image (6)

in which

Image (7)

is an integrating factor of the ordinary differential Eq. (2), and in which the stability angle small phi, Greek . has been defined such that:

Image (8)

The branch -90°<tan -1 <+90° is chosen for gravitationally stable stratifications, and the branch | tan -1 |>90° is chosen for gravitationally unstable stratifications.

The stability angle curly or open small phi, Greek . is a good descriptor of diffusive stability. For gravitationally stable stratifications, diffusive layering occurs when 45°<curly or open small phi, Greek <90° and salt fingering when -90°<curly or open small phi, Greek < -45°. For |curly or open small phi, Greek |<45°, the water column is diffusely stable. For |curly or open small phi, Greek |>90°, the water column is gravitationally unstable. The stability angle . is defined so that the stable range is centered on 0 (it differs by constant offsets from similar angles used by Ruddick (1983) and Washburn and Käse (1987). It has a geometric interpretation in the straight theta, small theta, Greek –s plane when the isopycnals are inclined at 45° with the s axis, as shown in Fig. 1.

If the `initial' condition is chosen so that small mu, Greek approximate 1 over the entire oceanic range of straight theta, small theta, Greek –s, then small pi, Greek will have the same physical dimensions as small rho, Greek , and the stability angle will be approximately related to the diapycnal gradient of spiciness:


Image (9)

2.3. Constructing spiciness

Spiciness small pi, Greek is constructed using potential density referenced to the surface pref=0 following Fofonoff (1985). The solution is expressed as


Image (10)

and the bij are found by minimizing the weighted mean square error on (1) and on small mu, Greek approximate 1 :

Image (11)

The integration is carried over the domain delimited by (0.32), (30.35), (30.38) and (0.35), containing most of the water masses of the oceans (seeWorthington and Levitus). The weights small lambda, Greek 1=0.8 and small lambda, Greek 2=0.2 were chosen to emphasize the geometric constraint between the spiciness isopleths and the isopycnals; the resulting coefficients are not very sensitive to this choice. The coefficients bij are given in Table 1; as a convention, b00=0 so that small pi, Greek (0,35)=0. The families of isopycnals and spiciness isopleths are shown in Fig. 2.

Table 1. Coefficients of the polynomial expansion (10). In the last column, (k) means .10k. (coefficient, c- and matlab programs available at http://satftp.soest.hawaii.edu/spice/)
View Table (<1K)


Enlarge Image (15K)

Fig. 2. The families of isopycnals (dashed) and spiciness isopleths (solid) with the reference pressure at the surface. With the scales chosen for this graph, the lines are orthogonal at about 8.5 °C. The domain over which the error (11) was minimized is delimited by dotted lines.

The rms and maximum errors on (1) are 2×10-3 and 5×10-3, and the rms and maximum errors on small mu, Greek approximate 1 are 1.4×10-2 and 4×10-2 respectively, the maximum values being confined to the edges of the integration domain. For the critical cases of the stability limits curly or open small phi, Greek =±45°, the corresponding rms and maximum errors on curly or open small phi, Greek are 0.4° and 1.2° are negligible for practical purposes.

3. Example

Fig. 3 shows a profile through a Mediterranean water eddy (`salt lens'), south of the Azores (Armi & Zenk, 1984). The corresponding straight theta, small theta, Greek –s curve is shown in Fig. 4. Fig. 5a shows the profile of spiciness using potential density as the ordinate. The isotherms and isohalines are overlaid. In this representation, they are approximately at ±45° with the small rho, Greek –small pi, Greek axes, and indicate the limits of diffusive stability: a segment of the profile steeper than an isotherm or an isohaline is diffusely stable. Fig. 5b shows the corresponding stability angle profile, computed over vertical intervals larger than the scale of the fine-structure observed below the salt lens.


Enlarge Image (10K)

Fig. 3. Vertical profiles through a Mediterranean salt lens, using pressure as the ordinate, taken on 17 September 1984 at 31°42'N; 22°00'W (courtesy of L. Armi). (a) salinity and potential temperature (referenced to the surface), (b) potential density and spiciness.


Enlarge Image (12K)

Fig. 4. straight theta, small theta, Greek –s curve for the profile shown in Fig. 3. The isopycnals and isospices are overlaid.


Enlarge Image (16K)

Fig. 5. (a) profile of spiciness using potential density as the ordinate; the profile is diffusely stable where the slope is steeper than the isotherms and isohalines, indicated by dashed lines. (b) corresponding profile for the stability angle curly or open small phi, Greek , computed by a least-square fit over 50 m intervals.

The main thermocline, from A to B, is slightly diffusively unstable to salt-fingering, withcurly or open small phi, Greek ~-72°(Rsmall rho, Greek ~2). A small diffusively-stable section BC is followed by the top of the salt lens CD, unstable to layering, with curly or open small phi, Greek ~-72°(Rsmall rho, Greek ~0.5). The core of the lens DE is diffusively stable. Below the lens, there is a region strongly unstable to salt-fingering, with curly or open small phi, Greek ~-84°(Rsmall rho, Greek ~1.25), associated with conspicuous thermohaline inversions and staircase-like structures.

In Fig. 5a, the stability angle is simply the angle between the tangent to the curve and the vertical (cf. Fig. 1 rotated 45° clockwise). This representation has thus the advantage of displaying the stability regime graphically, without the need to explicitly compute scale-dependent derivatives as in Fig. 5b.

4. Discussion

Analogous variables were proposed by Veronis and Jackett. Veronis (1972) constrained the isopleths of his variable small tau, Greek V (straight theta, small theta, Greek ,s) to be orthogonal to the isopycnals in the straight theta, small theta, Greek –s plane. However his definition depends on the scaling of the axes: geometric orthogonality is lost when the scales are altered, i.e. his constraint is invariant for rotations, but not for scale changes, which should be required for a state variable.

The variable small tau, Greek JMD introduced by Jackett and McDougall (1985) differs from small pi, Greek in three important aspects. Firstly, they enforce the constraint (1) exactly on a single spiciness isopleth, instead of in a least square sense over the domain of ocean water masses. Secondly, they require that, along all isopycnals, dsmall tau, Greek JMD=2small rho, Greek -1partial differential ssmall rho, Greek ds; although this is also related to double-diffusive convection (i.e. Ruddick & Turner, 1979), Eq. (9) provides a more transparent interpretation in terms of diffusive stability, with direct applications ( Ruddick, 1983). Thirdly, small tau, Greek JMD is dimensionless, whereas small pi, Greek has the same physical dimensions as density, yielding simple and elegant expressions for ((6), (7) and (9)). For practical purposes and over the range of oceanic water masses, linear conversions between small tau, Greek JMD and small pi, Greek can be obtained by:


small tau, Greek JMD=0.97small pi, Greek +23.8 (12a)

and

small pi, Greek =1.03small tau, Greek JMD-24.5 (12b)

with standard errors less than 0.025 in units of either small pi, Greek or small tau, Greek JMD.

5. Conclusions

Spiciness, as defined here, is ideally suited for the combined description of interleaving and double-diffusive processes at the boundary between different water masses, such as, for example, shelf-slope and coastal upwelling fronts, or marginal seas outflows. It has the practical advantage of lumping together the concept of a state variable decoupled from density, and the concept of an indicator of diffusive stability.

The small rho, Greek –small pi, Greek plane has an elegant geometrical property: isotherms and isohalines are approximately straight lines, the non-linearity of the equation of state being nearly entirely contained in their variable spacing. In other words, with properly engraved non-linear rulers for straight theta, small theta, Greek and for s, a straight theta, small theta, Greek –s diagram could be constructed entirely by hand, without recourse to a calculator or to special pre- printed isopycnal paper.

A word of caution: spiciness, just like potential density, is only useful over limited vertical excursions near the pressure to which they are referenced; for large vertical ranges, the slope of the isopycnals and spiciness isopleths vary significantly with pressure, and generalization of the polynomial expansion to include a reference pressure dependence is needed. The maximum pressure-induced slope error for spiciness between the surface and 1000 m depth, amounts to 1.5° at straight theta, small theta, Greek =20 and s=35, and 3° at straight theta, small theta, Greek =10 and s=35.


Acknowledgements

The author would like to thank an anonymous referee for his thorough review, which helped clarify an earlier version of this note. This work was supported by the Office of Naval Research. Contribution number 5323 of the School of Ocean and Earth Science and Technology.


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Corresponding author: Pierre Flament, Department of Oceanography, School of Ocean and Earth Sciences and Technology, University of Hawaii at Manoa, 1000 Pope Road, Honolulu, HI 96822, USA. Tel.: +1-808-956-6663; fax: +1-808-956-9225; email: pflament@soest.hawaii.edu