Chapter 1: INTRODUCTION
characteristic length scale. The transition from a laminar to turbulent flow
usually occurs at Re cr ~10
3 .
Convective instability of the flow results from unstable temperature or
salinity stratification. A relevant nondimensional criterion for thermal
convection is the Rayleigh number,
3 /(
)
T
T
Ra
g Th
D
NQ
'
,
(1.20)
where g is the acceleration of gravity, D T is the thermal expansion coefficient
of seawater (
4
2.6 10
T
D
o C
-1 at T = 20
o C and S = 35 psu), 'T is the
vertical temperature difference between the top and bottom of the convecting
layer, his the layer thickness, and N 7 is the molecular coefficient of thermal
diffusivity (
7
1.3 10
T
N
m
2 s
-1 at T = 20
o C and S = 35 psu). The term,
D T 'T = 'U/U, represents the fractional density difference between the top
and bottom of the convective layer. The transition from laminar to turbulent
convection is usually observed in the ocean for critical Rayleigh number Ra cr
~
4
5 1
u
The Reynolds number associated with ocean currents are very large, on
the order of 10
7 , and these currents are generally turbulent. Rayleigh
numbers in the ocean are also very large (typically greater than 10
14 for a
temperature difference of 0.1
o C over 10 m), so convection is usually
turbulent. However, for strong stable stratification Ra may drop below Ra cr .
In the near-surface ocean such cases can occur under extreme conditions of
calm weather and strong insolation or precipitation (Chapter 4).
transport of momentum and scalar properties like temperature, salinity, or
gas. In the case of a turbulent flow, coefficients K , K T , K S, and K C represent
the turbulent eddy transport. The conservation equations for momentum,
heat, salt, and passive tracers expressed in terms of turbulent eddy
coefficients appear exactly in the same form as their laminar analogs (1.1)(1.3) and (1.10)-(1.12). In a developed turbulent flow (i.e., at large Reynolds
K T , K S , and K C are approximately equal (unlike the
corresponding molecular diffusivities).
Turbulent eddy coefficients depend on the flow, geometry, and
stratification. In the simplest form, the turbulent eddy coefficient for
momentum is parameterized based on Prandtl’s mixing length hypothesis
~
K
lu and Kolmogorov’s hypothesis
~
l
u
b, where l is the mixing
length, and b is the turbulent kinetic energy (TKE). More sophisticated semiempirical closure schemes are considered in Chapters 3 and 5.
9
M
0 (Turner, 1973).
Although originally derived for laminar flows, parameterization laws
(1.5) and (1.13)-(1.15) are applicable for characterizing the turbulent
numbers), K ,
M
M
l
characteristic length scale. The transition from a laminar to turbulent flow
usually occurs at Re cr ~10
3 .
Convective instability of the flow results from unstable temperature or
salinity stratification. A relevant nondimensional criterion for thermal
convection is the Rayleigh number,
3 /(
)
T
T
Ra
g Th
D
NQ
'
,
(1.20)
where g is the acceleration of gravity, D T is the thermal expansion coefficient
of seawater (
4
2.6 10
T
D
o C
-1 at T = 20
o C and S = 35 psu), 'T is the
vertical temperature difference between the top and bottom of the convecting
layer, his the layer thickness, and N 7 is the molecular coefficient of thermal
diffusivity (
7
1.3 10
T
N
m
2 s
-1 at T = 20
o C and S = 35 psu). The term,
D T 'T = 'U/U, represents the fractional density difference between the top
and bottom of the convective layer. The transition from laminar to turbulent
convection is usually observed in the ocean for critical Rayleigh number Ra cr
~
4
5 1
u
The Reynolds number associated with ocean currents are very large, on
the order of 10
7 , and these currents are generally turbulent. Rayleigh
numbers in the ocean are also very large (typically greater than 10
14 for a
temperature difference of 0.1
o C over 10 m), so convection is usually
turbulent. However, for strong stable stratification Ra may drop below Ra cr .
In the near-surface ocean such cases can occur under extreme conditions of
calm weather and strong insolation or precipitation (Chapter 4).
transport of momentum and scalar properties like temperature, salinity, or
gas. In the case of a turbulent flow, coefficients K , K T , K S, and K C represent
the turbulent eddy transport. The conservation equations for momentum,
heat, salt, and passive tracers expressed in terms of turbulent eddy
coefficients appear exactly in the same form as their laminar analogs (1.1)(1.3) and (1.10)-(1.12). In a developed turbulent flow (i.e., at large Reynolds
K T , K S , and K C are approximately equal (unlike the
corresponding molecular diffusivities).
Turbulent eddy coefficients depend on the flow, geometry, and
stratification. In the simplest form, the turbulent eddy coefficient for
momentum is parameterized based on Prandtl’s mixing length hypothesis
~
K
lu and Kolmogorov’s hypothesis
~
l
u
b, where l is the mixing
length, and b is the turbulent kinetic energy (TKE). More sophisticated semiempirical closure schemes are considered in Chapters 3 and 5.
9
M
0 (Turner, 1973).
Although originally derived for laminar flows, parameterization laws
(1.5) and (1.13)-(1.15) are applicable for characterizing the turbulent
numbers), K ,
M
M
l