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Stability and Double Diffusion
5 Stability and Double Diffusion 5.1 Static stability Here we consider whether or not the variation of density with depth in the ocean is likely to cause the water to move vertically. If there is lightfluidon top of heavyfluidthen there will be no tendency for motion to occur. However, if there is heavyfluidabove lightfluidthere will be a tendency for the heavy fluid to sink and the light to rise—the density distribution is unstable. Thus we must examine the vertical density gradient to determine whether the fluid is stable, i.e. resists vertical motion, is neutral, i.e. offers no resistance to vertical motion, or is unstable, i.e. tends to move vertically of its own accord. If dp/dz < 0 (density increases with depth) we might expect thefluidto be statically stable so that if no motion is occurring the density distribution will not cause motion to occur. If dp/dz > 0, we expect the fluid to be unstable. When considering the density distribution in relation to stability we cannot ignore compressibility, i.e. the variation of density with pressure, which means with depth. In the case of neutral stability, if afluidparcel is moved up or down adiabatically (that is, with no heat exchange with its surroundings) and without salt exchange with its surroundings and then brought to a stop it will not tend to move further because wherever it is moved to it will have the same density as the surroundingfluid.Density must increase with depth because a parcel which is moved down will be compressed but must then have the same density as its surroundings. Thus in the neutral stability case, dp/dz < 0 and the fluid appears to be quite stable if the pressure effect is overlooked. At the same time, compression causes heating which decreases density, although not nearly enough to overcome the increase of density due to increased pressure and so, in the neutral stability case, temperature increases with increased depth, assuming that salinity effects can be ignored. If we neglect pressure effects we might think that the fluid is slightly unstable because we have colder fluid above warmer fluid. To take the compressibility into account one might try to consider the potential density or its anomaly, σθ, defined in Chapter 2. It is the density of the fluid when taken adiabatically to a reference pressure with the adiabatic temperature change taken into account. This procedure can be used in the 24
STABILITY AND DOUBLE DIFFUSION
25
atmosphere (provided that no condensation or evaporation occurs). In fact, an apparent or virtual potential temperature is used; this is the potential temperature which dry air would have if it had the same potential density as the moist air. Unfortunately, because of the complicated and non-linear equation of state for sea water, using potential density to determine the static stability does not always work in the ocean. For example, North Atlantic Deep Water has a slightly larger potential density than Antarctic Bottom Water. However, the former is found above, not below the latter. The temperature and salinity differences between these two water masses are sufficient that the variation of compressibility with these parameters leads to the in situ density (at the same depth) of the Antarctic Water being slightly greater than that of the North Atlantic Water, so that the Antarctic Water flows under the North Atlantic Water. The apparent instability in this case is caused by the fact that the reference pressure for σθ is taken at the surface (p = 0). If a reference pressure close to the in situ pressure is used then zero vertical variation of this potential density will indicate neutral stability. However, to consider the stability over the whole water column no single reference pressure is satisfactory in all cases and it is necessary to calculate a local value of stability as a function of depth as described below. 5.11
Criterion for static stability (E)
Suppose that the density of a stationary water mass changes with depth in some arbitrary manner and that at level 1 (Fig. 5.1), depth = — z, pressure = p, the in situ water properties are (p, S, T). Then a parcel of water is moved a short distance vertically from level 1 to level 2 without exchanging heat or salt with its surroundings. At level 2, the depth = — (z + δζ) and pressure = (p 4- <5p), and Z A SURROUNDING WATER
PARCEL LEVEL
©
(P)
*
(g) (z + 8z)U
FIG. 5.1
(W)
p,
S , T,
p\
S , T+ST, ρ+δρ
p
p,
S,
T, p
p + Sp
Water properties for calculations of stability.
26
INTRODUCTORY DYNAMICAL OCEANOGRAPHY
the surrounding water properties are (p2, S 2 , T2). The water properties of the parcel at level 2 will be (p\ S, Γ + δΤ) and its pressure = ρ + δρ. Here STis the adiabatic change of temperature due to change of pressure, i.e. δΤ = (dr/dp)ad δρ (where the subscript 'ad' = adiabatic). As δρ = — pgôz (refer to Appendix 1), δΤ — — (d7'/'dp)ad pgδζ = —Γδζ where Γ stands for the adiabatic temperature gradient. It is the change of temperature with depth caused by pressure change and is positive, i.e. compression causes the temperature to increase. At level 2 the restoring force on the parcel of volume δ V2 will be F = (buoyant upthrust — weight). By Archimedes' Principle, buoyant upthrust = weight of surrounding fluid displaced. Hence SV2p2g-ÔV2p'g
F= =
(5.1)
àV2g{p2-p')
and its acceleration if released will be ^
ÔV2g(p2-pf)
F
M
ÔV2p' dz
P
)
w
\di
\p dz
)p_
'-δζ)] (5.2)
where the subscript W refers to the surrounding water and the subscript P to the parcel. In equation (5.2) the change of density of the surrounding water
\d±6 1 =\ë±ôA
Idz Z\w and of the water parcel \_dz
\p
[_dSdz
|_
êT
+
ôTôz '
d dp dp AdZ
dpdz]
δζ
dp dz j P
because the salinity does not change as it (absolute salinity) is measured in grams/kilogram and is therefore independent of pressure effects. Now (dp/dz)w= (dp/dz)P, and if the changes of salinity and temperature between levels 1 and 2 are not large, then {dp/dp)w = (dp/dp)P because the δ&ρ and δίρ terms in the specific volume anomaly (equation (2.1)) are small and slowly varying. Also, (l/p)(dp/dz)öz in the denominator of equation (5.2) vanishes in the limit as δζ -* 0 and may be neglected. Then equation (5.2) becomes Q
^C-+r
dp_3S_ '5Γ + P\_dS~dz~ dT\dz
bz
(5.3)
STABILITY AND DOUBLE DIFFUSION
27
which is the ratio of the restoring acceleration of the displaced parcel to the acceleration due to gravity. Hesselberg defined the stability E of the water column as E= I
V
£
i.e.
I for δζ = unit length,
aJ
lTdpdS
+
dpidT
+r
\~l
=-pL^ ^U )J m
(5 4)
·
If E > 0, i.e. positive, the water is stable and a parcel displaced a short distance vertically will tend to return to its original position. Because it has inertia it will tend to overshoot its original position and then to oscillate about it, hence the stability of the water may be related to the occurrence of internal waves (Chapter 12). If E = 0, the water is neutrally stable and a displaced parcel will tend to remain in its displaced position. If E < 0, i.e. negative, the water will be unstable and a parcel which is displaced will tend to continue its displacement, i.e. overturn of the water should occur. 5.12
Numerical values for stability
In the open ocean, values of E in the upper 1000 m are of the order of lOOx 1 0 ~ 8 m _ 1 t o 1000 x 10 ~ 8 m ~ *, the largest values generally occurring in the upper few hundred metres. Below 1000 m depth, values decrease to less than 100 x 10" 8 m _ 1 and in deep trenches values close to 1 x 1 0 " 8 m _ 1 are found. In these latter cases, dS/dz is generally very small so that its effect on stability is negligible. Then as E -> 0 this means that dT/dz -► — Γ, i.e. the temperature change with depth in situ is close to the adiabatic rate due to change of pressure. The adiabatic rate increases from about 0.14°C/1000m at 5000 m to 0.19°C/1000m at 9000 m depth, the temperature changes being positive for increase of depth, i.e. the in situ temperature increases with depth in deep trenches. Note that in equation (5.4), dp/dS and dp/dT are taken holding the other variables fixed (Γ, p and 5, p respectively) at the local in situ values. This formula is not computationally very convenient because tables for density are not commonly available—it is the specific volume which is normally tabulated. To use such tables we use a = 1/p and hence (l/oc)(d(x/dS) = — (l/p)(dp/dS) and (1/α)(<3α/<3Γ) = — (\/p)(dp/dT). Making use of the expansion of a of equation (2.1) (omitting the <5(s, f, p) term which is negligible) equation (5.4) becomes
a|_ ds dz
ôT ôz
ds dz
δτ dz
\dT
Λ
ÔT )]
'
Thefirsttwo terms usually dominate and may be recognized as an expansion of (M s Jdz). The term involving Γ is generally quite small and may be ignored
28
INTRODUCTORY DYNAMICAL OCEANOGRAPHY
except in deep water where E is small. If E = 0, the neutral stability case, omitting the Γ term would give an apparent E of about — 2 x l 0 " 8 m - 1 near the surface and about — 4 x l 0 " 8 m " 1 at great depth, so the water would appear slightly unstable if the adiabatic temperature change with pressure (i.e. depth) were neglected as noted before. The importance of the other terms can be estimated by comparing them with the first two. The first and third terms have (l/
ÔT dz]
a dz ' 8
(
' '
_1
If the calculated values of £ are less than 50 x 10~ m , then the other terms should be included. The thermosteric anomaly Δ Μ is normally calculated from ot since they are directly related, as shown in Chapter 2, and at is usually calculated and tabulated along with 5 and T values during the first stage of data processing. Thus it is convenient to have an approximate formula for E in terms of at. Suppose that we expand the in situ density in a manner similar to that used for a (equation (2.1) as (5.7)
ρ-ΙΟΟΟ + σ, + β,,, + ε,,,
where a term of the form astp has been omitted because it will be negligible. Substituting this expansion into equation (5.4) and using datdS
gives
il
Js'dz'
'δσ, Jz
+
δε^ dS
δσ,δΤ δσ, ~δΤ~δζ= ~δζ~
ΐ*
de,,PdT δΤ δζ
The equivalent approximation to equation (5.6) is
E
--;£-
δτ J
(5.8)
59
<->
(5.9)
cz afirstapproximation to stability is From equations (5.6) and (5.9) we see pthat that Ast shall decrease with depth or that at shall increase with depth. Thus one can get an estimate of the sign of E just by looking at the tabulated values of
STABILITY AND DOUBLE DIFFUSION
29
these quantities. This is one of the reasons why at (or As t) is used rather than in situ values. (Another reason is that flow along constant at surfaces is easy since it is not restricted by static stability when equation (5.9) is a good approximation.) If one included the Γ term of equation (5.8) in equation (5.9) then it would essentially be equivalent to E = — (1/ρ)(δσθ/δζ). However, as we go a long way from the reference pressure (p = 0) the terms, other than the Γ term, not in the approximate equations (5.6) and (5.9) become more important. Neglect of them leads to the apparent instability between the North Atlantic Deep Water and the Antarctic Bottom Water mentioned earlier. Much of the effect of the pressure on the density cancelled out in deriving equation (5.4). Note also that the part of the pressure effect which cancelled is quite large. Suppose that we had just considered the gradient of in situ density. If the stability of the water were neutral, the in situ gradient must be the same as that for the water parcel. Now
dp\ p\dz)P but
so
=l(dp\
dp=_
p\dp)addz
fÔp\ \dpjàd
I — I = T^T where C is the speed of sound,
--(¥)
- £ * 400x10-m-
and as stated earlier using the in situ density gives a false impression of quite stable conditions when the stability is actually neutral! If one wishes to use in situ density, p(s, t, p), then to correct for compressibility the stability is given by X J_P__9_ £ = - Γpdz ^ - ^ 2C2·
(510)
To use this formulation for E one would use the equation of state (IES 80) directly to determine p and a suitable equation for C which is also a function of S, T and p. Although the water should be unstable and expected to turn over whenever E is negative, in practice it is not uncommon to find values of E = — 25 to — 50 x 1 0 ~ 8 m _ 1 i n the upper 50 m of the sea with indications that the stratification is stable. As already shown, the neglect of the adiabatic temperature gradient and the other terms which are in equations (5.5) and (5.8) but not in (5.6) and (5.9) cannot account for such observations. It may be that some of these cases are in fact associated with weak convection but the observations are not detailed enough to detect it. Such apparent unstable situations may also be due to observational errors. In practice, E is calculated using finite differences with observations from discrete levels. The error in a at observation may easily be 5 x 10 ~ 3 (see Appendix 3) and the error, Δσ,, in the difference between two
30
INTRODUCTORY DYNAMICAL OCEANOGRAPHY
levels could easily be 10 " 2. With a depth difference Δζ of 20 m, the error in E ~ ± (1/p)(AaJ Az)maybt AE ~ + 50 x 10~ 8 m -1 . At greater depths where Δζ is larger (because the difference between observation levels is usually greater) the errors will be smaller, e.g. for Az = 500 m and an error in Δσ, of 10 " 2, the error in £ i s only 2 x 10" 8 m _ 1 . Tables of values for dp/dS, dp/dT and Γ (as δθ/δζ) for the calculation of E using equation (5.4) are given in Neumann and Pierson (1966) or one may use tables of AStt, ôsp, ôtp in equation (5.5). 5.13 Buoyancy frequency (N) The Brunt-Väisälä (or buoyancy) frequency N is given by (radians s ^ 2 .
(5.11) /2
The frequency in cycles s "* (hertz) is Ν/2π = (g E γ /2π. It can be shown that this is the maximum frequency of internal waves in water of stability E. High values of N are usually found in the main pycnocline zone, i.e. where the vertical density gradient is greatest. This is usually in the thermocline in oceanic waters (where density variations are determined chiefly by temperature variations) or in the halocline in coastal waters (where density variations may be determined chiefly by salinity variations). 5.2 Double diffusion Even though the water column may be statically stable at a particular time, instability may develop because sea water is a multi-component fluid and the rates at which heat and salt diffuse molecularly are different. A result is that if two water masses of the same density but different combinations of temperature and salinity are in contact, one above the other, the differential ("double") diffusion of these two properties may give rise to density changes which render the layers unstable. This is an active area of research and reviews of the subject may be found in Turner (1973,1981). The details are beyond the scope of the present book but the general ideas are interesting and double diffusion may play a significant role in small-scale mixing in the oceans and in the formation of "fine" structure, the small scale (one to a few metres) vertical variations in temperature and salinity which have been found in the oceans as observations have improved with the use of continuously recording STD or CTD instruments (Salinity or Conductivity, Temperature, Depth). We consider the stability of cases with positive static stability but with no motion initially, because if there is motion, particularly turbulent motion generated by velocity shear or strong static instability, turbulent diffusion will
STABILITY AND DOUBLE DIFFUSION
31
dominate and probably prevent double-diffusion effects from becoming important. However, it seems that the ocean is sufficiently statically stable in some parts that shear generated turbulence is suppressed and double-diffusive effects may be important. Suppose that there is a layer of warmer, saltier water above cooler, fresher water, such that the upper layer is of the same density as or is less dense than the lower layer. Then the saltier water at the interface will lose heat to the cooler water below faster than it will lose salt because the rate of molecular diffusion of heat is about 100 times that for salt. If the density difference between the layers is small, the saltier water above may become heavier than the cooler, fresher layer below and sink downward into this layer. Likewise the cold, fresh water below the interface gains heat faster than salt and may become light enough to rise into the upper layer. The situation is referred to as one of "double-diffusive instability". The falling and rising motion occurs (in laboratory experiments later verified by field observations, Williams, 1975) in the form of thin columns and the phenomenon is called "salt fingering". There is evidence for its occurrence in the ocean at the lower surface of the outflow of warm, saline Mediterranean water from the Strait of Gibraltar into the cooler, less saline Atlantic water. If a layer of colder, fresher water is above a layer of warmer, saltier water, the water just above the interface becomes lighter than that above it and tends to rise while water below gets heavier and tends to sink. This phenomenon is called "layering" and may lead to fairly homogeneous layers separated by thinner regions of high gradients of temperature and salinity. There is evidence for its occurrence in the Arctic Ocean among other locations. Both of these processes could lead to the vertical transports of heat and salt being greater than the molecular diffusion rates, and to greater mixing than would occur if these processes were not possible. Of course, once the motion begins it may become dynamically unstable and break down into smaller scale turbulent motions and become very complicated. Dynamic instability is discussed briefly in the next section and also in Chapter 7. The final possibility of a warmer, fresher layer above a cooler, saltier layer does not allow a double-diffusive instability. The fresher water cools so that it does not tend to rise but it cannot get colder than the saltier water below and therefore does not tend to sink. Similarly, the saltier water does not tend to move up or down. For double diffusion to occur, the gradients of temperature and salinity across the interface must have the same sign; then, since they affect density oppositely, double diffusion may occur. 5.3
Dynamic stability
Even if the water is statically stable and double diffusion is not permitted by the temperature and salinity distributions, if motion is initiated it may be
32
INTRODUCTORY DYNAMICAL OCEANOGRAPHY
dynamically unstable and it may break down into smaller-sized irregular turbulent motions. This possibility will be discussed further after we have examined the equations of motion. Turbulent flows are familiar to everyone although they may not normally be labelled as such. Examples are the flow in most rivers, the gusty wind and the flow of water out of a tap, among many others. All these flows are very irregular both as a function of time at a fixed point and from point to point at a given time. The strong mixing caused by turbulent flow is often used, e.g. in stirring milk and sugar into coffee. After the stirring is stopped, the flow will gradually become more regular providing an example of non-turbulent flow, a type of flow which is less familiar in everyday experience.
STABILITY AND DOUBLE DIFFUSION
25
atmosphere (provided that no condensation or evaporation occurs). In fact, an apparent or virtual potential temperature is used; this is the potential temperature which dry air would have if it had the same potential density as the moist air. Unfortunately, because of the complicated and non-linear equation of state for sea water, using potential density to determine the static stability does not always work in the ocean. For example, North Atlantic Deep Water has a slightly larger potential density than Antarctic Bottom Water. However, the former is found above, not below the latter. The temperature and salinity differences between these two water masses are sufficient that the variation of compressibility with these parameters leads to the in situ density (at the same depth) of the Antarctic Water being slightly greater than that of the North Atlantic Water, so that the Antarctic Water flows under the North Atlantic Water. The apparent instability in this case is caused by the fact that the reference pressure for σθ is taken at the surface (p = 0). If a reference pressure close to the in situ pressure is used then zero vertical variation of this potential density will indicate neutral stability. However, to consider the stability over the whole water column no single reference pressure is satisfactory in all cases and it is necessary to calculate a local value of stability as a function of depth as described below. 5.11
Criterion for static stability (E)
Suppose that the density of a stationary water mass changes with depth in some arbitrary manner and that at level 1 (Fig. 5.1), depth = — z, pressure = p, the in situ water properties are (p, S, T). Then a parcel of water is moved a short distance vertically from level 1 to level 2 without exchanging heat or salt with its surroundings. At level 2, the depth = — (z + δζ) and pressure = (p 4- <5p), and Z A SURROUNDING WATER
PARCEL LEVEL
©
(P)
*
(g) (z + 8z)U
FIG. 5.1
(W)
p,
S , T,
p\
S , T+ST, ρ+δρ
p
p,
S,
T, p
p + Sp
Water properties for calculations of stability.
26
INTRODUCTORY DYNAMICAL OCEANOGRAPHY
the surrounding water properties are (p2, S 2 , T2). The water properties of the parcel at level 2 will be (p\ S, Γ + δΤ) and its pressure = ρ + δρ. Here STis the adiabatic change of temperature due to change of pressure, i.e. δΤ = (dr/dp)ad δρ (where the subscript 'ad' = adiabatic). As δρ = — pgôz (refer to Appendix 1), δΤ — — (d7'/'dp)ad pgδζ = —Γδζ where Γ stands for the adiabatic temperature gradient. It is the change of temperature with depth caused by pressure change and is positive, i.e. compression causes the temperature to increase. At level 2 the restoring force on the parcel of volume δ V2 will be F = (buoyant upthrust — weight). By Archimedes' Principle, buoyant upthrust = weight of surrounding fluid displaced. Hence SV2p2g-ÔV2p'g
F= =
(5.1)
àV2g{p2-p')
and its acceleration if released will be ^
ÔV2g(p2-pf)
F
M
ÔV2p' dz
P
)
w
\di
\p dz
)p_
'-δζ)] (5.2)
where the subscript W refers to the surrounding water and the subscript P to the parcel. In equation (5.2) the change of density of the surrounding water
\d±6 1 =\ë±ôA
Idz Z\w and of the water parcel \_dz
\p
[_dSdz
|_
êT
+
ôTôz '
d dp dp AdZ
dpdz]
δζ
dp dz j P
because the salinity does not change as it (absolute salinity) is measured in grams/kilogram and is therefore independent of pressure effects. Now (dp/dz)w= (dp/dz)P, and if the changes of salinity and temperature between levels 1 and 2 are not large, then {dp/dp)w = (dp/dp)P because the δ&ρ and δίρ terms in the specific volume anomaly (equation (2.1)) are small and slowly varying. Also, (l/p)(dp/dz)öz in the denominator of equation (5.2) vanishes in the limit as δζ -* 0 and may be neglected. Then equation (5.2) becomes Q
^C-+r
dp_3S_ '5Γ + P\_dS~dz~ dT\dz
bz
(5.3)
STABILITY AND DOUBLE DIFFUSION
27
which is the ratio of the restoring acceleration of the displaced parcel to the acceleration due to gravity. Hesselberg defined the stability E of the water column as E= I
V
£
i.e.
I for δζ = unit length,
aJ
lTdpdS
+
dpidT
+r
\~l
=-pL^ ^U )J m
(5 4)
·
If E > 0, i.e. positive, the water is stable and a parcel displaced a short distance vertically will tend to return to its original position. Because it has inertia it will tend to overshoot its original position and then to oscillate about it, hence the stability of the water may be related to the occurrence of internal waves (Chapter 12). If E = 0, the water is neutrally stable and a displaced parcel will tend to remain in its displaced position. If E < 0, i.e. negative, the water will be unstable and a parcel which is displaced will tend to continue its displacement, i.e. overturn of the water should occur. 5.12
Numerical values for stability
In the open ocean, values of E in the upper 1000 m are of the order of lOOx 1 0 ~ 8 m _ 1 t o 1000 x 10 ~ 8 m ~ *, the largest values generally occurring in the upper few hundred metres. Below 1000 m depth, values decrease to less than 100 x 10" 8 m _ 1 and in deep trenches values close to 1 x 1 0 " 8 m _ 1 are found. In these latter cases, dS/dz is generally very small so that its effect on stability is negligible. Then as E -> 0 this means that dT/dz -► — Γ, i.e. the temperature change with depth in situ is close to the adiabatic rate due to change of pressure. The adiabatic rate increases from about 0.14°C/1000m at 5000 m to 0.19°C/1000m at 9000 m depth, the temperature changes being positive for increase of depth, i.e. the in situ temperature increases with depth in deep trenches. Note that in equation (5.4), dp/dS and dp/dT are taken holding the other variables fixed (Γ, p and 5, p respectively) at the local in situ values. This formula is not computationally very convenient because tables for density are not commonly available—it is the specific volume which is normally tabulated. To use such tables we use a = 1/p and hence (l/oc)(d(x/dS) = — (l/p)(dp/dS) and (1/α)(<3α/<3Γ) = — (\/p)(dp/dT). Making use of the expansion of a of equation (2.1) (omitting the <5(s, f, p) term which is negligible) equation (5.4) becomes
a|_ ds dz
ôT ôz
ds dz
δτ dz
\dT
Λ
ÔT )]
'
Thefirsttwo terms usually dominate and may be recognized as an expansion of (M s Jdz). The term involving Γ is generally quite small and may be ignored
28
INTRODUCTORY DYNAMICAL OCEANOGRAPHY
except in deep water where E is small. If E = 0, the neutral stability case, omitting the Γ term would give an apparent E of about — 2 x l 0 " 8 m - 1 near the surface and about — 4 x l 0 " 8 m " 1 at great depth, so the water would appear slightly unstable if the adiabatic temperature change with pressure (i.e. depth) were neglected as noted before. The importance of the other terms can be estimated by comparing them with the first two. The first and third terms have (l/
ÔT dz]
a dz ' 8
(
' '
_1
If the calculated values of £ are less than 50 x 10~ m , then the other terms should be included. The thermosteric anomaly Δ Μ is normally calculated from ot since they are directly related, as shown in Chapter 2, and at is usually calculated and tabulated along with 5 and T values during the first stage of data processing. Thus it is convenient to have an approximate formula for E in terms of at. Suppose that we expand the in situ density in a manner similar to that used for a (equation (2.1) as (5.7)
ρ-ΙΟΟΟ + σ, + β,,, + ε,,,
where a term of the form astp has been omitted because it will be negligible. Substituting this expansion into equation (5.4) and using datdS
gives
il
Js'dz'
'δσ, Jz
+
δε^ dS
δσ,δΤ δσ, ~δΤ~δζ= ~δζ~
ΐ*
de,,PdT δΤ δζ
The equivalent approximation to equation (5.6) is
E
--;£-
δτ J
(5.8)
59
<->
(5.9)
cz afirstapproximation to stability is From equations (5.6) and (5.9) we see pthat that Ast shall decrease with depth or that at shall increase with depth. Thus one can get an estimate of the sign of E just by looking at the tabulated values of
STABILITY AND DOUBLE DIFFUSION
29
these quantities. This is one of the reasons why at (or As t) is used rather than in situ values. (Another reason is that flow along constant at surfaces is easy since it is not restricted by static stability when equation (5.9) is a good approximation.) If one included the Γ term of equation (5.8) in equation (5.9) then it would essentially be equivalent to E = — (1/ρ)(δσθ/δζ). However, as we go a long way from the reference pressure (p = 0) the terms, other than the Γ term, not in the approximate equations (5.6) and (5.9) become more important. Neglect of them leads to the apparent instability between the North Atlantic Deep Water and the Antarctic Bottom Water mentioned earlier. Much of the effect of the pressure on the density cancelled out in deriving equation (5.4). Note also that the part of the pressure effect which cancelled is quite large. Suppose that we had just considered the gradient of in situ density. If the stability of the water were neutral, the in situ gradient must be the same as that for the water parcel. Now
dp\ p\dz)P but
so
=l(dp\
dp=_
p\dp)addz
fÔp\ \dpjàd
I — I = T^T where C is the speed of sound,
--(¥)
- £ * 400x10-m-
and as stated earlier using the in situ density gives a false impression of quite stable conditions when the stability is actually neutral! If one wishes to use in situ density, p(s, t, p), then to correct for compressibility the stability is given by X J_P__9_ £ = - Γpdz ^ - ^ 2C2·
(510)
To use this formulation for E one would use the equation of state (IES 80) directly to determine p and a suitable equation for C which is also a function of S, T and p. Although the water should be unstable and expected to turn over whenever E is negative, in practice it is not uncommon to find values of E = — 25 to — 50 x 1 0 ~ 8 m _ 1 i n the upper 50 m of the sea with indications that the stratification is stable. As already shown, the neglect of the adiabatic temperature gradient and the other terms which are in equations (5.5) and (5.8) but not in (5.6) and (5.9) cannot account for such observations. It may be that some of these cases are in fact associated with weak convection but the observations are not detailed enough to detect it. Such apparent unstable situations may also be due to observational errors. In practice, E is calculated using finite differences with observations from discrete levels. The error in a at observation may easily be 5 x 10 ~ 3 (see Appendix 3) and the error, Δσ,, in the difference between two
30
INTRODUCTORY DYNAMICAL OCEANOGRAPHY
levels could easily be 10 " 2. With a depth difference Δζ of 20 m, the error in E ~ ± (1/p)(AaJ Az)maybt AE ~ + 50 x 10~ 8 m -1 . At greater depths where Δζ is larger (because the difference between observation levels is usually greater) the errors will be smaller, e.g. for Az = 500 m and an error in Δσ, of 10 " 2, the error in £ i s only 2 x 10" 8 m _ 1 . Tables of values for dp/dS, dp/dT and Γ (as δθ/δζ) for the calculation of E using equation (5.4) are given in Neumann and Pierson (1966) or one may use tables of AStt, ôsp, ôtp in equation (5.5). 5.13 Buoyancy frequency (N) The Brunt-Väisälä (or buoyancy) frequency N is given by (radians s ^ 2 .
(5.11) /2
The frequency in cycles s "* (hertz) is Ν/2π = (g E γ /2π. It can be shown that this is the maximum frequency of internal waves in water of stability E. High values of N are usually found in the main pycnocline zone, i.e. where the vertical density gradient is greatest. This is usually in the thermocline in oceanic waters (where density variations are determined chiefly by temperature variations) or in the halocline in coastal waters (where density variations may be determined chiefly by salinity variations). 5.2 Double diffusion Even though the water column may be statically stable at a particular time, instability may develop because sea water is a multi-component fluid and the rates at which heat and salt diffuse molecularly are different. A result is that if two water masses of the same density but different combinations of temperature and salinity are in contact, one above the other, the differential ("double") diffusion of these two properties may give rise to density changes which render the layers unstable. This is an active area of research and reviews of the subject may be found in Turner (1973,1981). The details are beyond the scope of the present book but the general ideas are interesting and double diffusion may play a significant role in small-scale mixing in the oceans and in the formation of "fine" structure, the small scale (one to a few metres) vertical variations in temperature and salinity which have been found in the oceans as observations have improved with the use of continuously recording STD or CTD instruments (Salinity or Conductivity, Temperature, Depth). We consider the stability of cases with positive static stability but with no motion initially, because if there is motion, particularly turbulent motion generated by velocity shear or strong static instability, turbulent diffusion will
STABILITY AND DOUBLE DIFFUSION
31
dominate and probably prevent double-diffusion effects from becoming important. However, it seems that the ocean is sufficiently statically stable in some parts that shear generated turbulence is suppressed and double-diffusive effects may be important. Suppose that there is a layer of warmer, saltier water above cooler, fresher water, such that the upper layer is of the same density as or is less dense than the lower layer. Then the saltier water at the interface will lose heat to the cooler water below faster than it will lose salt because the rate of molecular diffusion of heat is about 100 times that for salt. If the density difference between the layers is small, the saltier water above may become heavier than the cooler, fresher layer below and sink downward into this layer. Likewise the cold, fresh water below the interface gains heat faster than salt and may become light enough to rise into the upper layer. The situation is referred to as one of "double-diffusive instability". The falling and rising motion occurs (in laboratory experiments later verified by field observations, Williams, 1975) in the form of thin columns and the phenomenon is called "salt fingering". There is evidence for its occurrence in the ocean at the lower surface of the outflow of warm, saline Mediterranean water from the Strait of Gibraltar into the cooler, less saline Atlantic water. If a layer of colder, fresher water is above a layer of warmer, saltier water, the water just above the interface becomes lighter than that above it and tends to rise while water below gets heavier and tends to sink. This phenomenon is called "layering" and may lead to fairly homogeneous layers separated by thinner regions of high gradients of temperature and salinity. There is evidence for its occurrence in the Arctic Ocean among other locations. Both of these processes could lead to the vertical transports of heat and salt being greater than the molecular diffusion rates, and to greater mixing than would occur if these processes were not possible. Of course, once the motion begins it may become dynamically unstable and break down into smaller scale turbulent motions and become very complicated. Dynamic instability is discussed briefly in the next section and also in Chapter 7. The final possibility of a warmer, fresher layer above a cooler, saltier layer does not allow a double-diffusive instability. The fresher water cools so that it does not tend to rise but it cannot get colder than the saltier water below and therefore does not tend to sink. Similarly, the saltier water does not tend to move up or down. For double diffusion to occur, the gradients of temperature and salinity across the interface must have the same sign; then, since they affect density oppositely, double diffusion may occur. 5.3
Dynamic stability
Even if the water is statically stable and double diffusion is not permitted by the temperature and salinity distributions, if motion is initiated it may be
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INTRODUCTORY DYNAMICAL OCEANOGRAPHY
dynamically unstable and it may break down into smaller-sized irregular turbulent motions. This possibility will be discussed further after we have examined the equations of motion. Turbulent flows are familiar to everyone although they may not normally be labelled as such. Examples are the flow in most rivers, the gusty wind and the flow of water out of a tap, among many others. All these flows are very irregular both as a function of time at a fixed point and from point to point at a given time. The strong mixing caused by turbulent flow is often used, e.g. in stirring milk and sugar into coffee. After the stirring is stopped, the flow will gradually become more regular providing an example of non-turbulent flow, a type of flow which is less familiar in everyday experience.