3.18 Double Diffusion
73
3.18.3 Double-Diffusive Layering
Suppose there is a cooler, fresher layer above warmer, saltier water with the same
density (Fig. 3.40b). In this case, molecular diffusion of heat warms the surface
layer from below. As a result of this, convection develops in the top layer. Similarly,
cooling from above triggers convection in the bottom layer. In contrast to the doublediffusive instability, there is only little mixing across the interface. Moreover, the
molecular heat flux at the interface leads to the formation of a density difference
between the layers. The final result is a layer structure with a sharp density contrast
across the interface. This situation is referred to as double-diffusive layering.
3.18.4 The Gradient Ratio and the Turner Angle
A gradient ratio can be defined by:
R ρ =
α∂T /∂z
β∂ S/∂z
(3.74)
where α is the thermal expansion coefficient, and β is the salinity coefficient in the
equation of state (Eq. 3.65). This ratio gives the relative contributions of thermal
gradients and saline gradients to the density stratification. For a two-layer fluid, the
latter relation can be written as:
R ρ =
αΔT
βΔS
(3.75)
where ΔT and ΔS are temperature and salinity contrasts between the layers. Since
the gradient ratio can attain infinite values, it is more convenient to use the so-called
Turner angle, which is defined by (Ruddick, 1983):
T u = arctan
R ρ − 1
R ρ + 1
(3.76)
The Turner angle is used for the classification of different dynamic mixing
regimes that can develop in the ocean (Fig. 3.41). Turner angles |T u| > 90
◦ characterise unstable density stratification supporting the onset of thermohaline free convection (see Sect. 3.14). Turner angles of ±90
◦ characterise situations of vanishing
density gradients, whereas Turner angles in a range between ±45
◦ and ±90
◦ can
lead to either double-diffusive instability or double-diffusive layering. Turner angles
less than ±45
◦ correspond to a stably stratified configuration that does not support
either of these processes.
73
3.18.3 Double-Diffusive Layering
Suppose there is a cooler, fresher layer above warmer, saltier water with the same
density (Fig. 3.40b). In this case, molecular diffusion of heat warms the surface
layer from below. As a result of this, convection develops in the top layer. Similarly,
cooling from above triggers convection in the bottom layer. In contrast to the doublediffusive instability, there is only little mixing across the interface. Moreover, the
molecular heat flux at the interface leads to the formation of a density difference
between the layers. The final result is a layer structure with a sharp density contrast
across the interface. This situation is referred to as double-diffusive layering.
3.18.4 The Gradient Ratio and the Turner Angle
A gradient ratio can be defined by:
R ρ =
α∂T /∂z
β∂ S/∂z
(3.74)
where α is the thermal expansion coefficient, and β is the salinity coefficient in the
equation of state (Eq. 3.65). This ratio gives the relative contributions of thermal
gradients and saline gradients to the density stratification. For a two-layer fluid, the
latter relation can be written as:
R ρ =
αΔT
βΔS
(3.75)
where ΔT and ΔS are temperature and salinity contrasts between the layers. Since
the gradient ratio can attain infinite values, it is more convenient to use the so-called
Turner angle, which is defined by (Ruddick, 1983):
T u = arctan
R ρ − 1
R ρ + 1
(3.76)
The Turner angle is used for the classification of different dynamic mixing
regimes that can develop in the ocean (Fig. 3.41). Turner angles |T u| > 90
◦ characterise unstable density stratification supporting the onset of thermohaline free convection (see Sect. 3.14). Turner angles of ±90
◦ characterise situations of vanishing
density gradients, whereas Turner angles in a range between ±45
◦ and ±90
◦ can
lead to either double-diffusive instability or double-diffusive layering. Turner angles
less than ±45
◦ correspond to a stably stratified configuration that does not support
either of these processes.