| Citation: | Yan Li, Fangli Qiao, Hongyu Ma, Qiuli Shao, Zhixin Zhang, Guansuo Wang. The mechanism of the banded structure of drifting macroalgae in the Yellow Sea[J]. Acta Oceanologica Sinica, 2021, 40(7): 31-41. doi: 10.1007/s13131-021-1771-9
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In summer 2008, a massive bloom of non-toxic green macroalgae occurred in the coastal waters of the Yellow Sea off Qingdao, the host city of the 2008 Olympic Games sailing competition. And the blooms have happened annually since then on. Most of the drifting macroalgae appeared in linear bands on the sea surface. The lengths of the individual bands ranged from hundreds of meters to tens of kilometers, while the distance between the neighboring bands was about 1 km, ranging from 0.5 km to 6.0 km (Qiao et al., 2009). The water depth at the regions where algal bands appeared is about 30 m. Details about the drifting macroalgae are shown in Fig. 1.
Irving Langmuir first noticed the banded structure of floating material on the ocean surface while crossing the North Atlantic in 1927; he described how floating material was drawn into lines of convergence between pairs of vortices aligned downwind (Langmuir, 1938). This phenomenon was named as Langmuir circulation. Langmuir reported that the distance between the bands of floating Sargassum was 100–200 m, while the windrows in Lake George in his follow-up experiment had a separation of 5–25 m. Subsequent researches such as Pollard (1977), Leibovich (1983), and Thorpe (2004) reported scales ranging from 2 m to a few hundred meters. It is amzing to notice that the distance between neighboring macroalgae bands is 0.5–6.0 km, which is far greater than what is expected in Langmuir circulation, the spatial scale is about one order than that of Langmuir circulation. After the corresponding author of this paper, Fangli Qiao, published the observation results (Qiao et al., 2009), Thorpe (2009) immediately noticed the results and suggested that particles floating on the sea surface are drawn into bands by the converging flows between neighboring Langmuir cells, and the floating bands, in particular, the buoyant algae, can subsequently amalgamate as a result of Langmuir turbulence. He developed an idealized model to describe the rearrangement and dispersion of the floating particles. Although Thorpe (2009) proposed the above model, Fangli Qiao has been thinking that the banded structure should be a novel dynamic process.
Dong (1997) investigated the formation mechanism of tidal sand ridges with neighboring ridges ranging from 1 km to 10 km. That scale is on the same order as the observed spatial scale of the algal band in the Yellow Sea. Dong (1997) suggested that the sand ridges may be caused by the instability of the vertical shear of the tidal currents. However, the vertical shear of tidal current is so weak in the upper ocean, so the banded structure should be irrelevant with the instability of tidal current.
In atmospheric sciences, bands of clouds, known as “cloud streets”, are also common phenomenon and have been dynamically investigated. The banded structure of cloud streets can be clearly seen in Fig. 2. Lilly (1966) explained cloud streets and enriched the theory of the atmospheric boundary layer. He suggested that cloud streets were caused by instability of the atmospheric boundary layer. Brown (1970) considered the effect of temperature stratification and concluded that the characteristic depth of eddies in the atmospheric and oceanic boundary layer is 5–7 times of the Ekman characteristic length, and the associated wavelength is 4π times of this parameter.
In order to reveal dynamic mechanism of the banded structure from the surface-drifting macroalgae on the sea surface in the Yellow Sea, we implemented the classical linear stability analysis which has been used in Dong (1997) to explain the tidal sand ridges, Lilly (1966) and Brown (1970) to explain the cloud street. We dynamically explained this interesting phenomenon based on stability analysis of the ocean currents in the surface layer. The ocean current is separated into the primary (or mean) flow and the secondary flow (disturbance). The lateral and vertical structure of the disturbance, as determined from the lateral wave-length and the vertical amplitude and phase relations, is determined by the most unstable perturbation of the solution to the first-order stability equation. The mean flow used in this paper is a set of observed ocean current profiles. Since vertical shear of mean flow is the driving force of the secondary flow, so we removed the tidal current which has no or quite weak vertical shear at the surface layer. By numerically solving the Orr-Sommerfeld equation, we obtained the growth rate of unstable perturbations for different wave numbers and deflection angles from the mean flow. The most unstable perturbation is regarded as the dynamical process responsible for the banded structure. It is amazing that the spatial scale of the most unstable perturbation fits well with that of macroalgea bands, that is, about 1 km.
The governing equations comprise the equations of motion for a viscous, incompressible fluid in a rotating coordinate system and the continuity equations are as follows:
| $$\left\{\begin{aligned}& \dfrac{\partial u}{\partial x}+\dfrac{\partial v}{\partial y}+\dfrac{\partial w}{\partial z}=0, \\ & \dfrac{\partial u}{\partial t}+u\dfrac{\partial u}{\partial x}+v\dfrac{\partial u}{\partial y}+w\dfrac{\partial u}{\partial z}-fv=-\dfrac{1}{\rho }\dfrac{\partial p}{\partial x}+\\ & \quad {A}_{\rm{H}}\left(\dfrac{{\partial }^{\rm{2}}u}{\partial {x}^{2}}+\dfrac{{\partial }^{\rm{2}}u}{\partial {y}^{2}}\right)+{A}_{\rm{V}}\dfrac{{\partial }^{\rm{2}}u}{\partial {z}^{2}} \dfrac{\partial v}{\partial y},\\ &\dfrac{\partial v}{\partial t}+u\dfrac{\partial v}{\partial x}+v\dfrac{\partial v}{\partial y}+ w\dfrac{\partial v}{\partial z}+fu=-\dfrac{1}{\rho }\dfrac{\partial p}{\partial y}+ \\ &\quad {A}_{\rm{H}}\left(\dfrac{{\partial }^{\rm{2}}v}{\partial {x}^{2}}+\dfrac{{\partial }^{\rm{2}}v}{\partial {y}^{2}}\right)+{A}_{\rm{V}}\dfrac{{\partial }^{\rm{2}}v}{\partial {z}^{2}}, \\ &\dfrac{\partial w}{\partial t}+u\dfrac{\partial w}{\partial x}+v\dfrac{\partial w}{\partial y}+w\dfrac{\partial w}{\partial z}=-\dfrac{1}{\rho }\dfrac{\partial p}{\partial z}-g+\\ &\quad\; {A}_{\rm{H}}\left(\dfrac{{\partial }^{\rm{2}}w}{\partial {x}^{2}}+\dfrac{{\partial }^{\rm{2}}w}{\partial {y}^{2}}\right)+{A}_{\rm{V}}\dfrac{{\partial }^{\rm{2}}w}{\partial {z}^{2}},\end{aligned} \right.$$ | (1) |
where u and v are the horizontal velocity components in the x and y directions, respectively;
The boundary conditions are
| $$\left\{\begin{aligned} &u = v = w = 0, \quad z = - H;\\ &\dfrac{{\partial u}}{{\partial z}} = \dfrac{{\partial v}}{{\partial z}} = 0,\quad w = 0,\quad z = 0;\end{aligned}\right.$$ | (2) |
where H is the water depth. To obtain the perturbation equations, we divide
| $$\left\{\begin{aligned} &u = \overline u + {u^*},\\ & v = \overline v + {v^*},\\ & w = \overline w +{ w^*},\\ & p = \overline p + {p^*}.\end{aligned}\right.$$ | (3) |
Upon substituting Eq. (3) into Eq. (1), we get the full equations:
| $$\begin{cases} & \left(\dfrac{{\partial \overline u }}{{\partial x}} + \dfrac{{\partial \overline v }}{{\partial y}} + \dfrac{{\partial \overline w }}{{\partial z}}\right) + \left(\dfrac{{\partial u^*}}{{\partial x}} + \dfrac{{\partial v^*}}{{\partial y}} + \dfrac{{\partial w^*}}{{\partial z}}\right) = 0,\\ & \left(\dfrac{{\partial \overline u }}{{\partial t}} + \dfrac{{\partial u^*}}{{\partial t}}\right) + \left(\overline u \dfrac{{\partial \overline u }}{{\partial x}} + \overline u \dfrac{{\partial u^*}}{{\partial x}} + u^*\dfrac{{\partial \overline u }}{{\partial x}} + u^*\dfrac{{\partial u^*}}{{\partial x}}\right) +\\ & \quad\; \left(\overline v \dfrac{{\partial \overline u }}{{\partial y}} + \overline v \dfrac{{\partial u^*}}{{\partial y}} + v^*\dfrac{{\partial \overline u }}{{\partial y}} + v^*\dfrac{{\partial u^*}}{{\partial y}}\right) + \\ & \quad \;\left(\overline w \dfrac{{\partial \overline u }}{{\partial z}} + \overline w \dfrac{{\partial u^*}}{{\partial z}} + w^*\dfrac{{\partial \overline u }}{{\partial z}} + w^*\dfrac{{\partial u^*}}{{\partial z}}\right) - f\overline v - fv^* \\ &\quad\; = - \dfrac{1}{\rho }\dfrac{{\partial \overline p }}{{\partial x}} - \dfrac{1}{\rho }\dfrac{{\partial p^*}}{{\partial x}} + {A_{\rm{H}}}\left( {\dfrac{{{\partial ^{\rm{2}}}\overline u }}{{\partial {x^2}}} + \dfrac{{{\partial ^{\rm{2}}}\overline u }}{{\partial {y^2}}}} \right) + {A_{\rm{V}}}\dfrac{{{\partial ^{\rm{2}}}\overline u }}{{\partial {z^2}}}{\rm{ + }}\\ &\quad\;{A_{\rm{H}}}\left( {\dfrac{{{\partial ^{\rm{2}}}u^*}}{{\partial {x^2}}} + \dfrac{{{\partial ^{\rm{2}}}u^*}}{{\partial {y^2}}}} \right) + {A_{\rm{V}}}\dfrac{{{\partial ^{\rm{2}}}u^*}}{{\partial {z^2}}},\\ &\left(\dfrac{{\partial \overline v }}{{\partial t}} + \dfrac{{\partial v^*}}{{\partial t}}\right) + \left(\overline u \dfrac{{\partial \overline v }}{{\partial x}} + \overline u \dfrac{{\partial v^*}}{{\partial x}} + u^*\dfrac{{\partial \overline v }}{{\partial x}} + u^*\dfrac{{\partial v^*}}{{\partial x}}\right) +\\ & \quad\;\left(\overline v \dfrac{{\partial \overline v }}{{\partial y}} + \overline v \dfrac{{\partial v^*}}{{\partial y}} + v^*\dfrac{{\partial \overline v }}{{\partial y}} + v^*\dfrac{{\partial v^*}}{{\partial y}}\right)+\\ & \quad\;\left(\overline w \dfrac{{\partial \overline v }}{{\partial z}} + \overline w \dfrac{{\partial v^*}}{{\partial z}} + w^*\dfrac{{\partial \overline v }}{{\partial z}} + w^*\dfrac{{\partial v^*}}{{\partial z}}\right) + f\overline u + fu^*\\ & \quad\;= - \dfrac{1}{\rho }\dfrac{{\partial \overline p }}{{\partial x}} - \dfrac{1}{\rho }\dfrac{{\partial p^*}}{{\partial x}} + {A_{\rm{H}}}\left( {\dfrac{{{\partial ^{\rm{2}}}\overline v }}{{\partial {x^2}}} + \dfrac{{{\partial ^{\rm{2}}}\overline v }}{{\partial {y^2}}}} \right) + {A_{\rm{V}}}\dfrac{{{\partial ^{\rm{2}}}\overline v }}{{\partial {z^2}}}{\rm{ + }}\\ &\quad\;{A_{\rm{H}}}\left( {\dfrac{{{\partial ^{\rm{2}}}v^*}}{{\partial {x^2}}} + \dfrac{{{\partial ^{\rm{2}}}v^*}}{{\partial {y^2}}}} \right) + {A_{\rm{V}}}\dfrac{{{\partial ^{\rm{2}}}v^*}}{{\partial {z^2}}},\\ & \left(\dfrac{{\partial \overline w }}{{\partial t}} + \dfrac{{\partial w^*}}{{\partial t}}\right) + \left(\overline u \dfrac{{\partial \overline w }}{{\partial x}} + \overline u \dfrac{{\partial w^*}}{{\partial x}} + u^*\dfrac{{\partial \overline w }}{{\partial x}} + u^*\dfrac{{\partial w^*}}{{\partial x}}\right) + \\ &\quad\;\left(\overline v \dfrac{{\partial \overline w }}{{\partial y}} + \overline v \dfrac{{\partial w^*}}{{\partial y}} + v^*\dfrac{{\partial \overline w }}{{\partial y}} + v^*\dfrac{{\partial w^*}}{{\partial y}}\right) + \\ &\quad\;\left(\overline w \dfrac{{\partial \overline w }}{{\partial z}} + \overline w \dfrac{{\partial w^*}}{{\partial z}} + w^*\dfrac{{\partial \overline w }}{{\partial z}} + w^*\dfrac{{\partial w^*}}{{\partial z}}\right)\\ &\quad\; = - \dfrac{1}{\rho }\dfrac{{\partial \overline p }}{{\partial x}} - \dfrac{1}{\rho }\dfrac{{\partial p^*}}{{\partial x}} - g + {A_{\rm{H}}}\left( {\dfrac{{{\partial ^{\rm{2}}}\overline w }}{{\partial {x^2}}} + \dfrac{{{\partial ^{\rm{2}}}\overline w }}{{\partial {y^2}}}} \right) + \\ &\quad\; {A_{\rm{V}}}\dfrac{{{\partial ^{\rm{2}}}\overline w }}{{\partial {z^2}}}{\rm{ + }}{A_{\rm{H}}}\left( {\dfrac{{{\partial ^{\rm{2}}}w^*}}{{\partial {x^2}}} + \dfrac{{{\partial ^{\rm{2}}}w^*}}{{\partial {y^2}}}} \right) + {A_{\rm{V}}}\dfrac{{{\partial ^{\rm{2}}}w^*}}{{\partial {z^2}}}. \end{cases}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!$$ | (4) |
Because
| $$\left\{ \begin{aligned} & \dfrac{\partial u^*}{\partial x}+\dfrac{\partial v^*}{\partial y}+\dfrac{\partial w^*}{\partial z}=0,\\ & \dfrac{\partial u^*}{\partial t}+\overline{u}\dfrac{\partial u^*}{\partial x}+u^*\dfrac{\partial \overline{u}}{\partial x}+\overline{v}\dfrac{\partial u^*}{\partial y}+v^*\dfrac{\partial \overline{u}}{\partial y}+w^*\dfrac{\partial \overline{u}}{\partial z}-\\ & \quad\;fv^* =-\dfrac{1}{\rho }\dfrac{\partial p^*}{\partial x}+ {A}_{\rm{H}}\left(\dfrac{{\partial }^{\rm{2}}u^*}{\partial {x}^{2}}+\dfrac{{\partial }^{\rm{2}}u^*}{\partial {y}^{2}}\right)+{A}_{\rm{V}}\dfrac{{\partial }^{\rm{2}}u^*} {\partial {z}^{2}}, \\ &\dfrac{\partial v^*}{\partial t}+\overline{u}\dfrac{\partial v^*}{\partial x}+u^*\dfrac{\partial \overline{v}}{\partial x}+ \overline{v}\dfrac{\partial v^*}{\partial y}+v^*\dfrac{\partial \overline{v}}{\partial y}+w^*\dfrac{\partial \overline{v}}{\partial z}+\\ & \quad\;fu^* =-\dfrac{1}{\rho }\dfrac{\partial p^*}{\partial y}+{A}_{\rm{H}}\left(\dfrac{{\partial }^{\rm{2}}v^*}{\partial {x}^{2}}+\dfrac{{\partial }^{\rm{2}}v^*}{\partial {y}^{2}}\right)+ {A}_{\rm{V}}\dfrac{{\partial }^{\rm{2}}v^*}{\partial {z}^{2}},\\ &\dfrac{\partial w^*}{\partial t}+\overline{u}\dfrac{\partial w^*}{\partial x}+\overline{v}\dfrac{\partial w^*}{\partial y}=-\dfrac{1}{\rho }\dfrac{\partial p^*}{\partial z}+\\ &\quad\;{A}_{\rm{H}}\left(\dfrac{{\partial }^{\rm{2}}w^*}{\partial {x}^{2}}+\dfrac{{\partial }^{\rm{2}}w^*}{\partial {y}^{2}}\right)+{A}_{\rm{V}}\dfrac{{\partial }^{\rm{2}}w^*}{\partial {z}^{2}}.\end{aligned}\right.$$ | (5) |
The boundary conditions for the perturbation equations are
| $$\begin{cases} &u^* = v^* = w^* = 0,\quad z = - {{H}};\\ &\dfrac{{\partial u^*}}{{\partial z}} = \dfrac{{\partial v^*}}{{\partial z}},\quad w^* = 0,\quad z = 0.\end{cases}$$ | (6) |
Since the macroalgae mainly gathered in the region of (35.0°–37.0°N, 119.2°–122.0°E) in the Yellow Sea, we collected in-situ ocean current data from one station located at (36°02′N, 121°05′E). The position of the observation site is shown in Fig. 3. The data were obtained with an acoustic Doppler current profiler. We processed the data as follows: (1) used a 7-point smoothing filter, any velocity signal with period less than 1 h was removed; (2) the one-hour interval data were selected; and (3) low-pass filtering was performed using the Lanczos window cosine squared filter, and the mean flow removing the main tidal current was obtained. In fact, this mean flow is the wind driven Ekman drift current.
Figure 4 shows the obtained one-hour interval current data from 14:00 to 19:00 on 16 July. There is no significant difference in the velocity structure, so we just show the three-dimensional vector diagram of the velocity at 14:00 in Fig. 5.
For many cases of the banded structure, there is usually an angle
| $$ \begin{cases} & x' = {{(x\cos \theta + y\sin \theta)} / D} ,\\ & y' = {{(- x\sin \theta + y\cos \theta)} / D}, \\ & z' = {z / D}, \\ & u' = {{(u^*\cos \theta + v^*\sin \theta)} / {{u_0}}}, \\ & v' = {{(- u^*\sin \theta + v^*\cos \theta)} / {{u_0}}}, \\ & w' = {{w^*} / {{u_0}}} ,\\ & u = {{({\bar{{u}}}\cos \theta + \bar v\sin \theta)} / {{u_0}}}, \\ & v = {{(- \bar u\sin \theta + \bar v\cos \theta)} / {{u_0}}}.\\ \end{cases} $$ | (7) |
Since the secondary circulation is a two-dimensional flow in the new coordination system, each variable of the disturbance is uniform in the direction x, so the derivative along the direction x is zero. We can obtain the non-dimensional perturbation equations as follows:
| $$\tag{8-1}\frac{{\partial v'}}{{\partial y'}} + \frac{{\partial w'}}{{\partial z'}} = 0,$$ |
| $$\tag{8-2}{{Re}} \left(\frac{{\partial u'}}{{\partial t}} + v\frac{{\partial u'}}{{\partial y'}} + w'\frac{{\partial u}}{{\partial z'}}\right) - 2v' = \text{γ} \frac{{{\partial ^2}u'}}{{\partial y{'^2}}} + \frac{{{\partial ^2}u'}}{{\partial z{'^2}}},$$ |
| $$\tag{8-3}{{Re}} \left(\frac{{\partial v'}}{{\partial t}} + v\frac{{\partial v'}}{{\partial y'}} + w'\frac{{\partial v}}{{\partial z'}} + \frac{{\partial p'}}{{\partial y}}\right) + 2u' = \text{γ} \frac{{{\partial ^2}v'}}{{\partial y{'^2}}} + \frac{{{\partial ^2}v'}}{{\partial z{'^2}}},$$ |
| $$\tag{8-4}{{Re}} \left(\frac{{\partial w'}}{{\partial t}} + v\frac{{\partial w'}}{{\partial y'}} + \frac{{\partial p'}}{{\partial z}}\right) = \text{γ} \frac{{{\partial ^2}w'}}{{\partial y{'^2}}} + \frac{{{\partial ^2}w'}}{{\partial z{'^2}}},$$ |
where
The two-dimensional continuity equation in Eq. (8-1) allows the introduction of a stream function
By cross differentiation of Eqs (8–3) and (8–4), a new equation can be obtained:
| $$\begin{split} & {\mathop{{Re}}\nolimits} \left(\frac{\partial }{{\partial t}}{\nabla ^2}\psi ' + v\frac{\partial }{{\partial y'}}{\nabla ^2}\psi ' - \frac{{{\partial ^2}\psi '}}{{\partial y'\partial z'}}\frac{{\partial v}}{{\partial z'}} - \frac{{\partial \psi '}}{{\partial y'}}\frac{{{\partial ^2}v}}{{\partial z{'^2}}}\right) - {\rm{2}}\frac{{\partial u'}}{{\partial z'}}\\ & = \beta \frac{{{\partial ^{\rm{4}}}\psi '}}{{\partial y{'^{\rm{4}}}}} + \left( {{\rm{1 + }}\text{γ} } \right)\frac{{{\partial ^{\rm{4}}}\psi '}}{{\partial y{'^2}\partial z{'^2}}} + \frac{{{\partial ^{\rm{4}}}\psi '}}{{\partial z{'^{\rm{4}}}}}.\\[-16pt] \end{split}$$ | (9) |
The purpose of the stability analysis was to investigate the evolution of the solution to Eqs (8-2) and (9) over time. When
Because the coefficients of Eqs (8-2) and (9) are related to z only, we assume the form of the disturbance to be
| $$\begin{cases} & u' = \hat u(z)\exp [{\rm{i}}(ay - \omega t)], \\ & \psi ' = \hat \psi (z)\exp [{\rm{i}}(ay - \omega t)], \end{cases} $$ | (10) |
where
| $$ \begin{cases} & \dfrac{{{{\rm{d}}^2}\hat u}}{{{\rm{d}}z{'^2}}} - \text{γ} {a^2}\hat u - {\rm{i}}a{\mathop{{Re}}\nolimits} \left[(v - {\omega / a})\hat u + \dfrac{{{\rm{d}}u}}{{{\rm{d}}z'}}\hat \psi \right] - {\rm{2}}\dfrac{{{\rm{d}}\hat \psi }}{{{\rm{d}}z'}} = 0,\\ & \dfrac{{{{\rm{d}}^4}\hat \psi }}{{{\rm{d}}z{'^4}}} - \left( {{\rm{1 + }}\text{γ} } \right){a^2}\dfrac{{{{\rm{d}}^2}\hat \psi }}{{{\rm{d}}z{'^2}}} + \beta {a^4}\hat \psi - {\rm{i}}a{\mathop{{Re}}\nolimits} \Biggr[(v - {\omega / a})\times\\ &\quad\quad\;\left(\dfrac{{{{\rm{d}}^2}\hat \psi }}{{{\rm{d}}z{'^2}}} - {a^2}\hat \psi \right) - \dfrac{{{{\rm{d}}^2}v}}{{{\rm{d}}z{'^2}}}\hat \psi \Bigg] + {\rm{2}}\dfrac{{{\rm{d}}\hat u}}{{{\rm{d}}z'}} = 0. \end{cases}$$ | (11) |
The boundary conditions are
| $$\left\{\begin{aligned}& \hat \psi = \dfrac{{{\rm{d}}\hat \psi }}{{{\rm{d}}z'}} = \hat u = 0,\quad z' \!\to\! - {{{H}} / {{D}}};\\ &\hat \psi = \dfrac{{{{\rm{d}}^2}\hat \psi }}{{{\rm{d}}z{'^2}}} = \dfrac{{{\rm{d}}\hat u}}{{{\rm{d}}z'}} = 0,\quad z' = 0.\end{aligned}\right.$$ | (12) |
Because Eq. (11) and the boundary condition Eq. (12) are homogeneous, if it has nonzero solutions, the parameters
| $$ F({{Re}},a,\theta,\omega, \text{γ})=0.$$ | (13) |
Equation (13) can be called a characteristic relation. From this equation, we get the complex eigenvalue
To get the values of the non-dimensional scales we used above, we refer to the empirical relationship (shown below) between the surface velocity
| $$\left\{\begin{aligned}&{u_0}{\rm{/}}W{\rm{ = }}0.012\;7/\sqrt {\sin \left| \varphi \right|} ,\\ &{\overset\frown{D}} {\rm{ = }}4.3W/(\sqrt {\sin \left| \varphi \right|}),\end{aligned}\right.$$ | (14) |
where the unit of
| $$D{\rm{ = }}{\overset\frown{D}} {\rm{/\pi = }}4.3W/({\rm{\pi }}\sqrt {\sin \left| \varphi \right|}).$$ | (15) |
Thus, if we have data for latitude
| $$\left\{\begin{aligned}&{u_0} \approx {\rm{0}}{\rm{.11\;m/s}},\\ &D \approx 12.1\;{\rm{m}}.\end{aligned}\right.$$ | (16) |
The value of
In the following section, we conduct a series of numerical experiments using different Reynolds numbers, wave numbers, and declination angles. Their values are shown in Table 1.
| Parameters | Values |
| Reynolds number (${{Re} }$) | 100, 150, 200, 250, 300, 350, 400, 450 |
| AH/AV ($\text{γ}$) | 1:10:1000 |
| Wave number ($a$) | 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, 2.0 |
| Declination angle ($\theta $)/(°) | –50, –45, –40, –35, –30, –25, –20, –15, –10, –5, 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50 |
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In Fig. 7, we show how the imaginary part
Then, we check the spatial scale of secondary circulation. Figure 7 shows that the wave number of the unstable state has a large range, which is concentrated between 0.5 and 1.0. Therefore, we can get the wavelength
Meanwhile, if we take Re = 200 as an example, we show the max value of the imaginary part of the complex eigenvalue with varying wave number (Fig. 8). It is clear from the first panel of the figure that when the wave number value equals 0.05 or higher, the imaginary part of the complex eigenvalue becomes positive. The unstable state appears; the corresponding declination angle is about 5°. From the other panels, it is clear that with increasing wave number, the unstable state always exists until wave number value becomes 1.5, and the corresponding declination angle also increases gradually. Figure 9 shows the maximum value of the imaginary part of the complex eigenvalue with varying declination angle. When the declination angle equals 5° or higher, the imaginary part of the complex eigenvalue becomes positive, with the increase of the declination angle unstable state always existing until the wave number value equals 20°.
To consider the influence of the horizontal and vertical eddy viscosity coefficients
To go into further detail, when
To detect the character of the secondary circulation more clearly, we chose one unstable state for simulation. The choosed parameters are as follows: Re = 200, wave number (a) = 0.1, angle (
The stream function of the secondary circulation are shown in Fig. 12. The distance between two convergence fields is about 30 times of the ocean depth, so it is about 1 km. Therefore, it matches the observed spatial scale of the surface drifting macroalgae distribution, suggesting that secondary circulation related to the instability of the wind driven Ekman drift current is responsible for the banded structure.
The banded structure observed in the macroalgae bloom period in the Yellow Sea should indicate a new dynamic process in the upper ocean. From instability analysis, we found that the secondary circulation related to the stability of the wind driven Ekman drift current is responsible for the banded structure. The places where the banded structure occurred are convergence zones caused by a pair of secondary circulations, so any floating materials such as macroalgea bloom, ocean debris, oil spills will be accumulated in these convergent zones. In fact, both the oil pollution in the Gulf of Mexico in 2010 and the Sanchi oil spill in 2018 showed similar banded structure, which is incorrectly attribute to the Langmuir circulation. By the way, the divergent zone with the same spatial scale will have higher biological productivity. So, this new circulation could have a great influence on the ocean environment and marine ecosystem. For instance, it could have practical applications for the prevention and control of marine disasters, such as algal blooms, ocean debris and oil spills. More importantly, this is a new channel between the upper ocean and subsurface layer to exchange energy, materials, gas, etc. and this secondary circulation can be applied in the development of ocean and climate models. This new discovered secondary circulation, on which the corresponding author of this paper has paid high attention since 2008, in the ocean is a subject of high research value, which can enrich our knowledge on the ocean dynamics in the surface boundary layer.
By analyzing the featured distribution of the growth rates, we found that the spatial scale of the secondary circulation fits well with the observation by Qiao et al. (2009). This paper gives a dynamical explanation for the banded structure of the drifting macroalgae through traditional stability analysis. This result is not only an interesting phenomenon, but also may provide a new mechanism of vertical mixing in the upper ocean which is important for ocean and climate model development. The detailed research on the secondary circulation may be via observations; however, it is difficult to undertake such kind of direct measurement, which of course brings obstacles and difficulties for a designed scientific experiment. The macroalgea bloom provide us an excellent opportunity to reveal the mechanism of the secondary circulation.
| [1] |
Brown R A. 1970. A secondary flow model for the planetary boundary layer. Journal of the Atmospheric Sciences, 27(5): 742–757. doi: 10.1175/1520-0469(1970)027<0742:ASFMFT>2.0.CO;2
|
| [2] |
Dong Changming. 1997. Formation mechanism of modern tidal current sand ridges (in Chinese) [dissertation]. Qingdao: Institute of Oceanography, Chinese Academy of Sciences
|
| [3] |
Feng Shizuo, Li Fengqi, Li Shaojing. 1999. An Introduction to Marine Science (in Chinese). Beijing: Higher Education Press, 161
|
| [4] |
Langmuir I. 1938. Surface motion of water induced by wind. Science, 87(2250): 119–123. doi: 10.1126/science.87.2250.119
|
| [5] |
Large W G, McWilliams J C, Doney S C. 1994. Oceanic vertical mixing: a review and a model with a nonlocal boundary layer parameterization. Reviews of Geophysics, 32: 363–403. doi: 10.1029/94RG01872
|
| [6] |
Leibovich S. 1983. The form and dynamics of Langmuir circulation. Annual Review of Fluid Mechanics, 15: 391–427. doi: 10.1146/annurev.fl.15.010183.002135
|
| [7] |
Lilly D K. 1966. On the instability of Ekman boundary flow. Journal of the Atmospheric Sciences, 23(5): 481–494. doi: 10.1175/1520-0469(1966)023<0481:OTIOEB>2.0.CO;2
|
| [8] |
Noh Y, Min H S, Raasch S. 2004. Large eddy simulation of the ocean mixed layer: The effects of wave breaking and Langmuir circulation. Journal of Physical Oceanography, 34(4): 720–735. doi: 10.1175/1520-0485(2004)034<0720:LESOTO>2.0.CO;2
|
| [9] |
Pollard R T. 1977. Observations and theories of Langmuir circulations and their role in near surface mixing. In: Angel M, ed. A Voyage of Discovery: George Deacon 70th Anniversary Volume. Oxford: Pergamon, 235–251
|
| [10] |
Qiao Fangli, Dai Dejun, Simpson J, et al. 2009. Banded structure of drifting macroalgae. Marine Pollution Bulletin, 58(12): 1792–1795. doi: 10.1016/j.marpolbul.2009.08.006
|
| [11] |
Qiao Fangli, Yuan Yeli, Deng Jia, et al. 2016. Wave turbulence interaction induced vertical mixing and its effects in ocean and climate models. Philosophical Transactions of the Royal Society of London Series A—Mathematical, Physical and Engineering Sciences, 374(2065): 20150201. doi: 10.1098/rsta.2015.0201
|
| [12] |
Qiao Fangli, Yuan Yeli, Yang Yongzeng, et al. 2004. Wave-induced mixing in the upper ocean: distribution and application to a global ocean circulation model. Geophysical Research Letters, 31(11): L11303
|
| [13] |
Thorpe S A. 2004. Langmuir circulation. Annual Review of Fluid Mechanics, 36: 55–79. doi: 10.1146/annurev.fluid.36.052203.071431
|
| [14] |
Thorpe S A. 2009. Spreading of floating particles by Langmuir circulation. Marine Pollution Bulletin, 58(12): 1787–1791. doi: 10.1016/j.marpolbul.2009.07.022
|
Figures(12) / Tables(1)
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Yan Li, Fangli Qiao, Hongyu Ma, Qiuli Shao, Zhixin Zhang, Guansuo Wang. The mechanism of the banded structure of drifting macroalgae in the Yellow Sea[J]. Acta Oceanologica Sinica, 2021, 40(7): 31-41. doi: 10.1007/s13131-021-1771-9
| Parameters | Values |
| Reynolds number (${{Re} }$) | 100, 150, 200, 250, 300, 350, 400, 450 |
| AH/AV ($\text{γ}$) | 1:10:1000 |
| Wave number ($a$) | 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, 2.0 |
| Declination angle ($\theta $)/(°) | –50, –45, –40, –35, –30, –25, –20, –15, –10, –5, 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50 |
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