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In terms of the cross-filament wind and wave fields, a more specific 2D (x–z) filament is considered in this study, and the approximate wavy TTW balance equations suitable for LES (Sullivan and McWilliams, 2019) are as follows:
where
$ \left\langle{\cdot }\right\rangle $ represents the horizontal average along the down-filament axis (y-direction), x is the cross-filament direction, y is the down-filament direction, z is the vertical direction, f is the Coriolis parameter,$ \left\langle{v}\right\rangle $ is the down-filament geostrophic velocity,$ \left\langle{u}\right\rangle $ is the cross-filament ageostrophic velocity,$ \left\langle{w}\right\rangle $ is the vertical velocity,$ \left\langle{b}\right\rangle $ is the buoyancy,$ \left\langle{u'w'+{\tau }_{13}}\right\rangle $ is the cross-filament vertical momentum flux (the prime represents the fluctuation quantity;$ \left\langle{u'w'}\right\rangle $ is the resolved vertical momentum flux;$ \left\langle{{\tau }_{13}}\right\rangle $ is the vertical momentum flux of the subgrid scale in the cross-filament direction),$ \left\langle{v'w'+{\tau }_{23}}\right\rangle $ is the down-filament vertical momentum flux ($ \left\langle{v'w'}\right\rangle $ is the resolved vertical momentum flux;$ \left\langle{{\tau }_{23}}\right\rangle $ is the vertical momentum flux of the subgrid scale in the down-filament direction),$ {u}_{\mathrm{s}} $ is the Stokes drift velocity,$ \left\langle{{\omega }_{y}}\right\rangle $ is the down-filament vorticity, and$ \left\langle{{\omega }_{z}}\right\rangle $ is the vertical vorticity. Therefore, the magnitude of the Coriolis parameter ($ f $ ), the horizontal buoyancy gradient ($ \partial \left\langle{b}\right\rangle/\partial x $ ), the vertical gradient of the vertical momentum fluxes ($ \partial \left\langle{u'w'+{\tau }_{13}}\right\rangle/\partial z $ and$ \partial \left\langle{v'w'+{\tau }_{23}}\right\rangle/\partial z $ ), the Stokes-Coriolis force ($ {-fu}_{\mathrm{s}} $ inducing the anti-Stokes Euler current$ u=-{u}_{\mathrm{s}} $ ), and the composite vortex forces$ ({u}_{\mathrm{s}}\left(-\left\langle{{\omega }_{z}}\right\rangle+\partial \left\langle{{\omega }_{y}}\right\rangle/\partial x\right) $ ) directly impact the structures of submesoscale currents based on the wavy TTW relation (Suzuki et al., 2016; McWilliams, 2018; Sullivan and McWilliams 2019). In this LES model, the simulations are only driven by the cross-filament wind and wave fields and the cold filament is set in the buoyancy field. -
The dynamics of the upper mixed layer, including Langmuir turbulence and submesoscale currents, is assumed to be described by conventional wave-averaged equations (Hamlington et al., 2014; Haney et al., 2015; Suzuki et al., 2016; Sullivan and McWilliams, 2019). The LES equation set suiting for this simulation is given as follows:
where
$ {x}_{i} $ (i = 1, 2, 3) are the Cartesian coordinates,$ {u}_{i} $ are the resolved velocity components in the$ {x}_{i} $ directions,$ t $ is the time,$ {f}_{k} $ is the Coriolis parameter,$ {\omega }_{k} $ is the vorticity component,$ {u}_{\mathrm{s}j} $ is the Stokes drift velocity,$ {P}_{\mathrm{m}}=P/{\rho }_{{\mathrm{o}}}+2e/3+ 1/2\left[{\left({u}_{i}+{u}_{\mathrm{s}i}\right)}^{2}-{u}_{i}{u}_{i}\right] $ is the modified pressure, P is the pressure, e is the turbulent kinetic energy of the subgrid scale,$ {\tau }_{ij}={\nu }_{t}{S}_{ij} $ is the momentum flux of the sub grid scale,$ {\nu }_{tur} $ is the turbulent eddy viscosity,$ {S}_{ij}=1/2\left(\partial {u}_{i}/\partial {x}_{j}+\partial {u}_{j}/\partial {x}_{i}\right) $ is the strain tensor of the resolved velocities,$ {\xi }_{ijk} $ is the standard antisymmetric tensor,$ {\delta }_{i3} $ is the Kronecker delta,$ \rho =\rho \left[1-\alpha \left(T-{T}_{{\mathrm{o}}}\right)\right] $ is density,$ {\rho }_{{\mathrm{o}}} $ = 1000 kg m−3 is the reference density,$ \alpha $ is the thermal expansion coefficient,$ g $ is the gravity acceleration,$ {S}_{\mathrm{s}\mathrm{g}\mathrm{s}}=-{\tau }_{ij}{S}_{ij} $ is the shear production of the subgrid scale,$ {B}_{\mathrm{s}\mathrm{g}\mathrm{s}}=g{\tau }_{Temk}/{T}_{{\mathrm{o}}} $ is the buoyancy production of the subgrid scale, To is the reference temperature,$ \varepsilon =0.93{e}^{3/2}/\Delta $ ($ \Delta =\sqrt[3]{\Delta x\Delta y\Delta z} $ , where$ \Delta x $ ,$ \Delta y $ ,$ \Delta z $ are the grid spacings) is the dissipation rate,$ {D}_{\mathrm{s}\mathrm{g}\mathrm{s}}= \partial \left(2{v}_{t} \partial e/ \partial {x}_{i}\right)/ \partial {x}_{i} $ is the diffusion production of the subgrid scale, T is the resolved temperature,$ {\tau }_{Temj}=-{\nu }_{Tem}\partial T/\partial {x}_{i} $ is the heat flux of the subgrid scale, and$ {\nu }_{Tem} $ is the turbulent eddy diffusivity. The terms$ {\nu }_{t}=0.1l{e}^{1/2} $ and$ {\nu }_{Tem}=\left(1+2l/\Delta \right){\nu }_{tur} $ were suggested by Moeng (1984) and Sullivan et al. (1994), where$ l=\Delta $ within the upper mixed layer and$ l=0.76e^{1/2}\left(\mathit{\mathrm{\mathit{g}}}\partial T/T\mathrm{_o}\partial z\right) $ in the thermocline.The sea surface friction velocity (
$ {u}_{*} $ ) is calculated based on the wind velocity ($ U_{\mathrm{a}} $ ) at z = 10 m above the sea surface (Liu et al., 1979), as follows:where
$ {\tau }_{{\mathrm{oa}}} $ is the wind stress of the sea surface,$ {\rho }_{{\mathrm{o}}} $ is the sea water density,$ {\rho }_{{\mathrm{a}}} $ is the air density, and$ {C}_{\mathrm{d}} $ is the drag coefficient.The Stokes drift velocity of a monochromatic surface wave (
$ {u}_{{\mathrm{s}}} $ ) in the deep-water region (Skyllingstad and Denbo, 1995; McWilliams et al., 1997; Sullivan and McWilliams, 2019) is given bywhere
$ {u}_{{\mathrm{os}}} $ is the Stokes drift velocity of the sea surface,$ {\delta }_{{\mathrm{s}}}=1/2k $ is the Stokes depth scale,$ k=2\pi /\lambda $ is the wave number of a monochromatic surface wave, and$ \lambda $ is the wavelength. In this simulation, the direction of the wind and wave fields is oriented in the cross-filament (x) direction. In addition, misaligned wind and wave fields also widely appear in the ocean (Sullivan et al., 2012; Van Roekel et al., 2012; McWilliams et al., 2014). The unaligned wind and wave fields can enhance the complexities for the impact of Langmuir turbulence on the submesoscale currents of a cold filament. The influence of Langmuir turbulence with the unaligned wind and wave fields on the submesoscale currents of a cold filament will not be studied in this paper. The turbulent Langmuir number$ {{\mathrm{La}}}_{{\mathrm{tur}}}=\sqrt{{u}_{*}/{u}_{\mathrm{o}\mathrm{s}}} $ is the turbulent Langmuir number (McWilliams et al., 1997; Li et al., 2005). There is no heat flux at the sea surface. -
The 2D (x–z) model of the idealized dense filament is set in the temperature field of the simulation domain (McWilliams et al., 2015; Sullivan and McWilliams, 2019). The density
$ \rho ={\rho }_{{\mathrm{o}}}\left[1-\alpha \left(T-{T}_{{\mathrm{o}}}\right)\right] $ and buoyancy$ b=g\left({\rho }_{{\mathrm{o}}}- \rho \right)/{\rho }_{\mathrm{{o}}} $ are calculated by the temperature (T). The dense filament axis is aligned with the y direction. Hence, x, y and z are referred to as the cross-filament, down-filament, and vertical directions. The simulation domain is (Lx, Ly, H) = (12, 4.5, −0.25) km and the grid number is (Nx, Ny, Nz) = (4096, 1536, 256). The horizontal grid spacing is$ \Delta x $ =$ \Delta y $ = 2.93 m and the vertical grid spacing is$ \Delta z $ = 0.98 m. The grid scale in this study can resolve both the boundary layer turbulence and submesoscale currents (Wang, 2001; Sullivan and Patton, 2011; Hamlington et al., 2014; Smith et al., 2016; Suzuki et al., 2016). The grid number is larger than 1.6 billion, which means that the simulation needs the great amount of calculation due to the non-hydrostatic approximation.The initial state of the horizontal and vertical structure of the idealized and 2D dense filament consists of three layers—that is, an intermediate blended layer connects the upper and lower layers (McWilliams et al., 2015; McWilliams, 2017; Sullivan and McWilliams, 2018). The upper layer is z > −ho, the lower layer is z < −h3, and the intermediate blended layer is −ho < z < −h3. The mean buoyancy field in the cross-filament (x–z) plane is
where bs(x) and b3 are the buoyancy distribution in the upper and lower layers, respectively. In the upper layer (z > −ho), the buoyancy field and upper mixed layer depth are given by
where
$ \delta {b}_{{\mathrm{o}}} $ is the buoyancy variation between the left and right sides of the dense filament compared to the buoyancy bso in the centerline of the cross-filament axis. The$ \delta {h}_{{\mathrm{o}}} $ is the increase in the upper mixed layer depth compared to the background value ho directly below the central cold core region. The horizontal width of the dense filament is defined by the scale$ {W}_{{\mathrm{d}}} $ . The value x = 0 is taken to be the centerline of the dense filament.In the lower layer (z < −h3), the buoyancy is given by
where
$N_{\rm{o}}^2 $ is the background stratification and H is the depth of the computational domain.The smooth blending function
$ F\left(\zeta \right) $ for the intermediate blended layer (−ho < z < −h3) is given byThe function a(x), satisfying a derivative continuity condition, is given by
The constants employed in the above formulae are
The above definitions create an initially dense filament in the cross-filament (x–z) plane. The width of the dense filament is 2
$ {W}_{d} $ = 4 km. The LES directly solves the temperature equation. Hence, for reference, the values of bso = 8.51 × 10−3 m s−2 and$ \delta {b}_{{\mathrm{o}}} $ = 0.785 × 10−3 m s−2 correspond to a surface temperature difference of Tso − To = 5.2 K and a temperature jump of$ \delta {T}_{{\mathrm{o}}} $ = 0.48 K. The reference temperature is To = 10°C and the expansion coefficient is$ \alpha $ = 1.668 × 10−4 K−1 (Sullivan and McWilliams, 2018).The direction of the wind and wave fields is aligned with the cross-axis direction of the cold filament (the positive x-direction), as shown in Fig. 1. The magnitude of the wind and wave fields for the different simulation cases are shown in Table 1.
Figure 1. Sketch map of the cold filament, submesoscale currents (black arrows) diagnosed by the TTW balance, and the direction of the wind and wave fields (blue arrows) relative to the cold filament axis.
Experiment $ {u}_{*}/fh $ $ {U}_{a} $ (m s−1) $ {u}_{*} $ (10−2 m s−1) $ {u}_{{\mathrm{os}}} $ (10−1 m s−1) $ {\delta }_{s} $ (m) $ {{\mathrm{La}}}_{{\mathrm{tur}}} $ $ h $ (m) $ f $ (10−4 s−1) Exp.1 0.76 5 0.57 0.63 2.13 0.3 60 1.25 Exp.2 1.07 7 0.80 0.88 4.15 0.3 60 1.25 Exp.3 1.37 9 1.03 1.14 6.92 0.3 60 1.25 Exp.4 1.71 11 1.28 1.42 10.32 0.3 60 1.25 Exp.5 2.09 13 1.57 1.74 14.42 0.3 60 1.25 Table 1. Parameters of the simulation experiments. In the table,
$ {u}_{*}/fh $ is a dimensionless parameter,$ U_{\mathrm{a}} $ is the wind velocity at z = 10 m above the sea surface,$ {u}_{*} $ is the sea surface friction velocity,$ {u}_{{\mathrm{os}}} $ is the surface Stokes drift velocity,$ {\delta }_{{\mathrm{s}}} $ is the Stokes depth,$ {{\mathrm{La}}}_{{\mathrm{tur}}} $ is the turbulent Langmuir number, h is the initial depth of the upper mixed layer, and$ f $ is the Coriolis parameter. -
At the sea surface (z = 0), the fluid is driven by the imposed wind and wave fields. The stress-free conditions are used on the bottom boundary (z = −H) (Haney et al., 2015). The periodic boundary conditions are used in the horizontal (x–y) planes. The spatial discretization is the pseudospectral method in the horizontal directions and the second-order finite differences in the vertical direction (Sullivan and McWilliams, 2019). The time integral is advanced by the third-order Runge–Kutta scheme.
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Firstly, the simulation is integrated for an inertial period t =
$ 2\pi /f $ ≈ 14 h with a horizontal homogeneous upper mixed layer, which generates the well-developed Langmuir turbulence. The initialization of the cold filament frontogenesis with the well-developed Langmuir turbulence may stabilize the large-scale meandering and long-term disintegration of the density filament induced by the baroclinic instabilities (Kaminski and Smyth, 2019; Sullivan and McWilliams, 2019). Secondly, an idealized 2D (x–z) dense filament is introduced into the last volume from the well-developed Langmuir turbulence field and the simulation is forward-integrated by about 1.2 h, and then the ageostrophic secondary circulation fully develops. Thirdly, an idealized 2D (x–z) dense filament is again introduced into the last volume of the fully developed ageostrophic secondary circulations to avoid the possible effect of the buoyancy field slumping in the former integral time period, which is similar to the previous method of Sullivan and McWilliams (2018). Finally, the subsequent integral time references this time of the third step as t = 0 h and the simulation of submesoscale currents impacted by Langmuir turbulence is forward-integrated by more than 30 h.
Experiment | $ {u}_{*}/fh $ | $ {U}_{a} $ (m s−1) | $ {u}_{*} $ (10−2 m s−1) | $ {u}_{{\mathrm{os}}} $ (10−1 m s−1) | $ {\delta }_{s} $ (m) | $ {{\mathrm{La}}}_{{\mathrm{tur}}} $ | $ h $ (m) | $ f $ (10−4 s−1) |
Exp.1 | 0.76 | 5 | 0.57 | 0.63 | 2.13 | 0.3 | 60 | 1.25 |
Exp.2 | 1.07 | 7 | 0.80 | 0.88 | 4.15 | 0.3 | 60 | 1.25 |
Exp.3 | 1.37 | 9 | 1.03 | 1.14 | 6.92 | 0.3 | 60 | 1.25 |
Exp.4 | 1.71 | 11 | 1.28 | 1.42 | 10.32 | 0.3 | 60 | 1.25 |
Exp.5 | 2.09 | 13 | 1.57 | 1.74 | 14.42 | 0.3 | 60 | 1.25 |