Mixing process on the northeast coast of Hokkaido in summer

Abstract

Direct measurements using a free-falling micro-profiler were conducted on the northeast coast of Hokkaido in the summer of 2007 to clarify the mixing process in the Soya Warm Current (SWC) region in terms of microstructure. The distribution of the Turner angle (Tu) showed that these regions have a high potential for double diffusive convection, but direct measurements of the turbulent dissipation rate (ε) and dissipation of temperature variance ($\chi_{T}$) did not necessarily correspond to each other in the SWC region, especially in the offshore front of SWC and farther offshore. The mixing efficiency indicated that, even though the Turner angle (Tu) indicated a high potential for double diffusive convection, turbulent mixing was the main contributor to the mixing process in this region, and double-diffusive convection only contributed partially and sparsely, especially in the boundary off SWC water. The bottom mixed layer (BML) is known to thicken off the SWC. The vertical diffusivity coefficient was enhanced near the bottom (10⁻⁴–10⁻³ m² s⁻¹) off the SWC, and these results support that turbulence near the bottom off the SWC contributed to the thickening of the BML.

Appendix: Derivations of \( \bf{K_{T}^{{}}} \), \( \bf{K_{S}^{{}}} \), and \( \varvec{\gamma_{d}^{{}}}\) in the salt-fingering (\( \varvec{\partial \overline{{{\kern 1pt} T}} /\partial {\kern 1pt} z > 0 }\) and 1 < \(\varvec{R_{\rho }^{{}}} \) < 100) and diffusive cases ( \(\varvec{\partial \overline{{{\kern 1pt} T}} /\partial {\kern 1pt} z < 0 }\) and 0 < \(\bf{R_{\varvec{\rho}}^{{}}}\) < 1)

The expressions for $K_{T}^{{}}$, $K_{S}^{{}}$, and $\gamma_{d}^{{}}$ are the same between salt-fingering and diffusive cases because they are derived from the same set of equations and definitions. The equations and definitions used in the derivation are Eq. (4) and

$$ 0 \approx - g\frac{{\overline{\rho 'w\,'} }}{\rho } - \varepsilon , $$

(10)

$$ 0 \approx - \overline{T'w'} \frac{{\partial \overline{T} }}{\partial z} - \frac{{\chi_{T} }}{2} , $$

(11)

$$ \overline{\rho 'w'} = - \rho \alpha \overline{T'w'} + \rho \beta \overline{S'w'} , $$

(12)

$$ \overline{T'w'} \equiv - K_{T}^{{}} \frac{{\partial \overline{T} }}{\partial z} , $$

(13)

$$ \overline{S'w'} \equiv - K_{S}^{{}} \frac{{\partial \overline{S} }}{\partial z} , $$

(14)

$$ R_{\rho } \equiv \frac{{\alpha {\kern 1pt} \partial \overline{T} /\partial z}}{{\beta {\kern 1pt} \partial \overline{S} /\partial z}} , $$

(15)

$$ \gamma_{d} \equiv \frac{{\beta {\kern 1pt} {\kern 1pt} \overline{S'w'} }}{{\alpha {\kern 1pt} {\kern 1pt} \overline{T'w'} }} \equiv \gamma_{s}^{ - 1} , $$

(16)

$$ N^{2} \equiv - \frac{g}{\rho }\frac{{\partial \overline{\rho } }}{\partial z} = \alpha \frac{{\partial \overline{T} }}{\partial z} - \beta \frac{{\partial \overline{S} }}{\partial z} , $$

(17)

where the superscript “D.D.” was omitted here. Equation (10) is the stationary kinetic energy equation, where the shear production term is neglected because mixing is driven by the release of potential energy in double diffusion and Eq. (11) is the stationary equation for temperature variance. Those assumptions hold for both salt-fingering and diffusive double diffusion. Equation (12) is a consequence of the definitions of α and β; Eqs. (13) and (14) are the definitions of $K_{T}^{{}}$ and $K_{S}^{{}}$; and the rest are definitions.

Eliminating $\overline{T'w'}$ between Eqs. (11) and (13) gives

$$ K_{T} = \frac{{\chi_{T} }}{{2({{\partial \overline{T} } \mathord{\left/ {\vphantom {{\partial \overline{T} } {\partial z}}} \right. \kern-\nulldelimiterspace} {\partial z}})^{2} }}. $$

(18)

Eliminating $\overline{S'w'}$, $\overline{T'w'}$, $\overline{\rho 'w'}$, ε and $N^{2}$ among Eqs. (4), (10), (12), (13), (14), and (17) yields

$$ K_{S} = \frac{{(\alpha {\kern 1pt} \partial \overline{T} /\partial z - \beta {\kern 1pt} \partial \overline{S} /\partial z)/\Upgamma + \alpha \partial \overline{T} /\partial z}}{{\beta \partial \overline{S} /\partial z}}K_{T}, $$

which can be written as

$$ K_{S} = \frac{{R_{\rho } - 1 + \Upgamma R_{\rho } }}{{\Upgamma R_{\rho } }}R_{\rho } K_{T} , $$

(19)

using the definition of $R_{\rho }$ Eq. (15).

Using Eqs. (13), (14), and (16) gives

$$ \gamma_{d} \equiv \frac{{\beta \overline{S'w'} }}{{\alpha \overline{T'w'} }} = \frac{{\beta K_{S} \partial \overline{S} /\partial z}}{{\alpha K_{T} \partial \overline{T} /\partial z}} = \frac{1}{{R_{\rho } }}\frac{{K_{S} }}{{K_{T} }}. $$

(20)

Therefore $\gamma_{d}$ can be written as Eq. (9), and then we get Eq. (7).