Biogenic mixing induced by intermediate Reynolds number swimming in stratified fluids

Abstract

We study fully resolved motion of interacting swimmers in density stratified fluids using an archetypal swimming model called “squirmer”. The intermediate Reynolds number regime is particularly important, because the vast majority of organisms in the aphotic ocean (i.e. regions that are 200 m beneath the sea surface) are small (mm-cm) and their motion is governed by the balance of inertial and viscous forces. Our study shows that the mixing efficiency and the diapycnal eddy diffusivity, a measure of vertical mass flux, within a suspension of squirmers increases with Reynolds number. The mixing efficiency is in the range of O(0.0001–0.04) when the swimming Reynolds number is in the range of O(0.1–100). The values of diapycnal eddy diffusivity and Cox number are two orders of magnitude larger for vertically swimming cells compared to horizontally swimming cells. For a suspension of squirmers in a decaying isotropic turbulence, we find that the diapycnal eddy diffusivity enhances due to the strong viscous dissipation generated by squirmers as well as the interaction of squirmers with the background turbulence.

Introduction

Heated and important discussions are rooted in identifying the sources required to sustain ocean mixing. The mechanical power needed to sustain ocean mixing is estimated, using the Levitus climatology, to be more than 2 trillion watts¹. Mixing across stratification requires a source of kinetic energy². Even though winds and tides are found to be two major mechanical sources for providing the kinetic energy required for ocean mixing, researchers believe that other important sources remain undiscovered³.

Mixing can be characterized by the upwelling of water column¹

where w is the vertical convective velocity along the z-direction, ρ is the density of the fluid and K_ρ is the diapycnal eddy diffusivity of density which is a measure of the vertical mass flux. Direct estimates of vertical flux at ocean thermoclines are about one order of magnitude smaller than the bulk’s average value⁴. It suggests that there exist undersampled ‘hot spots’ in the ocean interior at which mixing is much stronger than the values obtained from measurements of vertical flux.

In the ocean interior, the mixing is supplied by small scale mixing events. To date, internal wave breaking generated through winds and abysmal tidal flows are considered to be the major postulated contributors to the interior ocean mixing², however, the direct measurements of mixing in the mid ocean are unreliable⁴. There exists enormous biomass in the ocean interior, which could contribute to mixing. Particularly, a large population of zooplankton and pelagic organisms lives in the mesopelagic region (200–1000 m in depth)⁵. Intensive biological activities such as diel vertical migration occur in the mid ocean. The energy imported by biomass is immediately available in the mid ocean while winds and tidal flows import the energy at sea surfaces or topologically complex boundaries. A majority of this energy is dissipated as heat while propagating in the ocean^3,5. Therefore, it is important to examine whether the mixing induced by swimming organisms is important in the ocean interior.

The idea of biogenic mixing was first examined by Munk⁶. The estimated diffusivity using the mixing length argument appeared to be negligible at a large length scale (~300 m). However, direct measurements of mixing in the local ‘hot spots’ showed that biogenic mixing could be important. Schooling behavior of marine organisms in an aggregate can produce strong energy dissipation. Huntely and Zhou⁷ first calculated the kinetic energy produced by marine organisms and found the energy dissipation to be on the order of 10⁻⁵ W ⋅ kg⁻¹, which is 3 to 4 orders of magnitude larger than the average background turbulence in the ocean. Kunze and coworkers measured turbulent energy production in a dense population of euphausiids during the dusk at Sannich Inlet, British Columbia and found it to be on the order of 10⁻⁵ W ⋅ kg⁻¹⁸. On the other hand, Visser⁹ demonstrated low mixing efficiency in these circumstances because the length scale at which marine organisms import energy is much smaller than the Ozmidov buoyancy length scale in the ocean (~3–10 m); he argued that the turbulent kinetic energy of small organisms is mainly dissipated to heat prior to contributing to mixing.

Katija and Dabiri^10,11 suggested mixing via induced drift volume as an alternative mechanism to biogenic mixing: the fluid convects with the migrating body and the mixing will be determined by the drift volume. However, the problem remains unresolved since the drifted volume can re-stratify in marine environments and may not induce mixing. Biogenic mixing cannot be addressed without performing the detailed analysis of swimming organisms in stratified fluids, as it has been done in the recent debate. The present paper provides a fully resolved analysis of the flow generated in a suspension of swimmers interacting in a stratified fluid and quantifies the induced mixing.

Biogenic mixing in a stratified fluid is insignificant when relevant swimming Reynolds number is below unity. Wagner et al.¹² estimated the mixing efficiency for swimming microorganisms using the solution of point force singularities in stratified fluids given by Ardekani and Stocker¹³ and found the biogenic mixing to be minute when the inertial effects are neglected. Kunze¹⁴ estimated the diapycnal eddy diffusivity K_ρ through scaling analysis and found it to be negligible for Re = Ua/ν < 1, where U is the characteristic swimming velocity of the organism, a is the body size of the organism and ν is the kinematic viscosity of the fluid. In the aphotic ocean (i.e. regions that are 200 m beneath the sea surface), zooplankton are the most abundant organisms leading to vertical fluid transport^6,15 and their body size ranges from millimeters to centimeters. The corresponding swimming Reynolds number is in the range of Re ~ O(1–100). Recent experimental studies have shown that swimming in the inertial regime generates enhanced dissipation rate whose spatial scale exceeds organisms’ body size in the absence^15,16 and presence of the density stratification¹⁷. Therefore, it is important to examine the hydrodynamic interaction of swimming organisms in this inertial regime and to quantify the induced mixing. In this manuscript, we study fully resolved motion of interacting swimmers in density stratified fluids using an archetypal swimming model called “squirmer”. Further, we consider a suspension of squirmers in a decaying isotropic turbulence, resembling the scenario where marine organisms hydrodynamically interact in the presence of the background turbulence.

Numerical Framework

We performed a fully-resolved three dimensional direct numerical simulation of a suspension of squirmers. Although the squirmer model was originally proposed to study low Reynolds number swimming^18,19, it has been recently extended to the inertial regime^20,21,22. It effectively models the flow generated by a coordinated beating of cilia on the surface of organisms, such as ctenophora, living worldwide in marine environments²³. Their sizes range from millimeters to centimeters and their Reynolds number is on the order of O(10²). For a spherical squirmer, an axisymmetric tangential surface velocity is given as¹⁸

where θ is the polar angle measured from the swimming direction, β scales with aω/U, ω is the vorticity generated by the squirmer, U and a are the characteristic swimming velocity and the radius of the squirmer, respectively. For example, aω/U for the copepod nauplii (Temora longicornis) is about two in a swimming mode (ω ~ 70 s⁻¹, a ~ 300 μm, U ~ 10⁻² m ⋅ s⁻¹)²⁴. β is positive (negative) for the organisms generating trust in front of (behind) their body called puller (pusher). The puller brings fluid from front and back and expels it from the side, while pusher does the opposite. The flow generated by a squirmer in a stratified fluid strongly affects the density field and consequently, the squirmer’s motion²⁵. Here, we investigate the mixing induced by a suspension of squirmers in stratified fluids in the absence and presence of the background turbulence. We should note, however, that the squirmer is only a reduced-order model for the locomotion of organisms. This reduced-order squirmer model has been broadly employed to examine various aspects of swimming at low Reynolds number, such as hydrodynamic interaction of two organisms²⁶, swimming near a wall¹⁹, suspension dynamics²⁷ and optimal feeding²⁸.

The governing equations for an incompressible viscous fluid under Boussinesq approximation are given as

where u = (u, v, w) is the flow velocity, is the density perturbation from the linear background density profile , p is the disturbance pressure, μ is the dynamic viscosity of the fluid, is the gravitational acceleration, is the unit vector along the direction of gravity, T is the temperature and κ is the thermal diffusivity. The body force f in Equation (4) accounts for the hydrodynamic effect imposed by freely swimming squirmers²⁹. The density ρ and ρ^* can be written as ρ = ρ_f + (ρ_p − ρ₀)Φ and ρ^* = ρ₀ + (ρ_p − ρ₀)Φ, respectively, where ρ_f is the fluid density, ρ₀ is the volume-averaged density of the fluid and ρ_p is the squirmer density. The phase indicator parameter Φ is unity inside the squirmer and zero elsewhere in the computational domain. Equations (3, 4, 5) are derived from the conservation of mass, momentum and energy for an incompressible fluid, respectively. The density variation across thermocline occurs due to the vertical variation in temperature, , where β_T is the thermal expansion coefficient, is the temperature perturbation and is the linear background temperature. The diffusivity coefficients are assumed to be uniform and the same for the squirmer and the background fluid³⁰. In order to obtain a steady-state condition, we set , where and correspond to the thermal expansion coefficient of the squirmer and background fluid, respectively. Equation (5) can be rewritten in terms of temperature perturbation T′ as

The density of the background fluid linearly changes with temperature, which is an appropriate assumption for centimeter size organisms. The magnitude of the stratification can be characterized by the Brunt-Väisälä frequency, . Considering both mild and strong density gradients in the ocean, the value of N is around 10⁻⁴ − 0.3 s^−131,32. The dynamics of swimming in a stratified fluid can be characterized by three independent dimensionless parameters. The buoyancy effects can be characterized by Froude number Fr = U/(Na). Reynolds number Re = Ua/ν characterizes the ratio of inertial to viscous forces. The Prandtl number Pr = ν/κ measures the ratio of the momentum diffusivity ν to the thermal diffusivity κ and its value is about 7 for temperature stratified fluids. Equations (3), (4) and (6) are solved numerically, the details of which are given in the ‘Methods’ section. Unless otherwise stated, squirmers are force-free, torque-free and neutrally buoyant (ρ_p = ρ₀) and they are initially oriented along the gravitational direction. The simulations are continued till the entire system reaches quasi-steady state when both kinetic energy and temperature perturbation satisfy¹²

Results

We performed simulations for a swarm of pushers and pullers in the range of Reynolds number between 0.1 and 100 in a linearly stratified fluid in the absence and presence of background turbulence. The biogenic mixing process is quantified by calculating the mixing properties, such as mixing efficiency, diapycnal eddy diffusivity and Cox number.

Huntley and Zhou⁷ have provided the empirical relationship for the packing density of the organisms in the ocean. The volume fraction for escaping mode ϕ_e and cruising mode ϕ_c can be estimated as

The estimated volume fraction of organisms swimming at Re = 10 is ϕ_e = 4.6% and ϕ_c = 1.34%. The volume fraction of marine organisms at an intermediate Reynolds number rarely exceeds 4%⁷. In this study, unless otherwise stated, we consider a volume fraction of 4% to obtain an upper bound of the biogenic mixing. Hydrodynamic interactions at this volume fraction, corresponding to a semi-dilute regime (<10%), are mainly pairwise²⁷. Consequently, small number of squirmers can accurately capture the physics of the problem.

Mixing efficiency

For a swarm of squirmers in a linearly stratified fluid, the kinetic energy equation, in a quasi-steady state, is written as

where S represents the squirmers’ surfaces, n is the unit vector normal to the surface S, V is the entire fluid domain and E is the strain rate tensor. The term on the left hand side of equation (10) is the total energy input generated by squirmers. The first and second terms on the right hand side of equation (10) represent the viscous dissipation and the rate of creation of gravitational potential energy in the entire fluid domain, respectively. The energy input is generated due to the fluid stress σ at the surface of the squirmers. Consequently, the disturbance induced by the freely moving squirmers is the main source of mechanical energy input to the surrounding fluid. On the other hand, turbulent mixing imports mechanical energy through the turbulent production term. For example, in a shear turbulence, the turbulent production term is , where is the turbulent Reynolds stress tensor and ∂Ũ/∂x is the mean shear of the turbulence. The efficiency of turbulent mixing, also called the flux Richardson number, has been defined as the ratio of the rate of removal of energy by the buoyancy forces to the total turbulent energy production^33,34. The total energy input balances the viscous dissipation and the buoyancy flux of the fluid. The efficiency Γ for a biogenic mixing process is defined as

where overbar denotes the averaged quantities over the entire fluid domain. Wagner et al. utilized equation (11) to estimate the mixing efficiency induced by a single micron-size microorganism which is on the order of 10⁻⁸ in an ocean environment¹². The analytic solution was derived using solution of point force singularities in a stratified fluid¹³ for a visco-diffusive regime, where the inertial effects are neglected. For a suspension of squirmers in a linearly stratified fluid, we find that the mixing efficiency increases with inertia (Fig. 1a). For 10 < Re < 100 and |β| = 5, the mixing efficiency achieved by pushers is larger than pullers. The largest mixing efficiency produced by squirmers is (0.042 ± 0.0036) for Re = 100, β = 1 and Fr ~ 5.2 which is a strong stratification (deeper estuaries have a typical value of N ~ 0.1³², leading to a Froude number in the range of 5 < Fr < 30). This value is smaller than the mixing efficiency in a typical turbulent mixing event (Γ ~ 0.17)³³.

Figure 1

The effect of inertia on the biogenic mixing is quantified.

(a) Mixing efficiency generated by a swarm of squirmers is plotted for different Reynolds numbers; the horizontal dashed line represents the mixing efficiency for a typical turbulent mixing, Γ = 0.17. (b) the normalized diapycnal eddy diffusivity is plotted for different Reynolds numbers; the horizontal solid line represents the value of kinematic viscosity (momentum diffusivity); the horizontal dashed line represents the diapycnal eddy diffusivity caused by the internal wave breaking in the ocean¹⁴. (c) The normalized kinetic dissipation between the shear-driven turbulence³⁴ (black circles) and mixing generated by squirmers are compared. The orange square corresponds to our calculation of turbulent mixing. (d) The Cox number is plotted for different Reynolds numbers; the dotted dashed and long dashed lines correspond to the measured minimum and maximum values of Cox number at North Pacific thermocline³⁶, respectively. The data points are time-averaged values after the system has reached quasi-steady state; the error bars represent the temporal standard deviation of the time-averaged quantity. The volume fraction is ϕ = 4% and Fr = 5.29.

Diapycnal eddy diffusivity

The diapycnal eddy diffusivity of density is defined as^31,33

and quantifies the mixing due to vertical transport³⁴. Kunze¹⁴ showed that K_ρ scales as (K_ρ/ν) ~ γ₁Re²ϕ, where γ₁(<1) is the correlation coefficient and ϕ is the volume fraction of the organisms in an aggregate. The eddy diffusivity obtained from the present simulations agrees well with Kunze’s scaling for Re < 1 and β = 1 (blue line in Fig. 1b). The diapycnal eddy diffusivity for Re < 1 and β = ±5 (−5 and +5 shown with black and red lines, respectively in Fig. 1b) deviates from Kunze’s formula due to the importance of hydrodynamic interactions between squirmers with large |β | generating strong vorticity. Therefore, the scaling analysis cannot capture the biogenic mixing generated by a suspension of squirmers even at low values of Reynolds numbers. The diapycnal eddy diffusivity increases with inertia. For Re > 10 (Fig. 1b), pushers generate more effective mixing than pullers and the magnitude of diapycnal eddy diffusivity at Re = 50 and β = −5 exceeds that of the molecular diffusion. Noss and Lork³⁵ experimentally evaluated the vertical mass flux generated by a freely swimming Daphina (Re ~ 50, a ~ 0.5 mm, U ~ 1 cm ⋅ s⁻¹, ϕ ~ 1%) in a density stratified fluid and showed its value to be about 0.8 × 10⁻⁵ m² ⋅ s⁻¹ which is consistent with our numerical results for pushers swimming at Re = 50. At Re = 100 which resembles swimming of centimeter-sized organisms, the mixing is comparable to small scale turbulent mixing caused by internal wave breaking (10⁻⁵ m² ⋅ s⁻¹), a major contributor to the interior ocean mixing^2,14.

It would be illustrative to compare the values of diapycnal eddy diffusivity due to biogenic mixing to those generated by turbulent events. For shear turbulence in a stratified fluid, three distinctive regions can be identified based on the turbulent activity parameter ε/νN²³⁴, the ratio of the destabilizing effect of turbulent stirring to the stabilizing effects originated from the combination of buoyancy and viscosity. Here, ε = 2νE : E. Strong mixing occurs for large values of ε/νN². Region I in Fig. 1c corresponds to the decaying turbulence in a stratified fluid for ε/νN² < 7; region II corresponds to the stationary turbulence which occurs when 7 < ε/νN² < 100; region III is associated with growing turbulence which occurs for ε/νN² > 100³⁴. Figure 1c shows that the majority of data points representing the biogenic mixing induced by squirmers lie above the scaling curve for the turbulent mixing. The dissipation generated by the squirmers in a stratified fluid corresponds to regions II and III. The mixing generated by the squirmers in local hot spots can generate eddy diffusivities as strong as turbulent mixing, but it occurs at larger values of energy dissipation compared to turbulent mixing.

Temperature microstructure

To quantify the local temperature microstructure at thermocline, the Cox number^31,36 is used and is defined as

where is the mean temperature gradient. The Cox number provides a measure of the variance of the temperature gradient in the fluid. Gregg³⁶ has reported the seasonal Cox number to be between 1 and 290 at North Pacific thermocline. For the mixing generated by the squirmers at an intermediate Reynolds number, the temperature microstructure is generated through the combination of the squirmer’s locomotion and the thermal diffusion of the fluid. The Cox number in a swarm of squirmers can be as large as O(100) (see Fig. 1d).

Effect of swimming orientation

Unless otherwise stated, the squirmers are initialized to swim in the vertical direction; however, their swimming orientation evolves over time due to their hydrodynamic interaction, collision as well as the background turbulence because they are modeled as force-free, torque-free swimmers. We have performed an additional simulation where the swimmers are constrained to swim in the vertical direction by applying an external torque. In this case, the mixing efficiency, diapycnal eddy diffusivity and Cox number enhance for vertically swimming pushers (β = −5, ϕ = 4%, Re =50, Fr = 5.29) by a factor of 1.72, 1.88 and 3.00, respectively, compared to force-free, torque-free swimmers. The mixing efficiency, diapycnal eddy diffusivity and Cox number enhance by a factor of 7.43, 7.98 and 8.26 for pullers (β = 5, ϕ = 4%, Re =50, Fr = 5.29), respectively.

The mixing induced by squirmers is sensitive to the initial swimming orientation α, measured from the gravitational direction. For a regular array of squirmers, the ratio of mixing efficiency for vertically swimming cells (α = 0°) to horizontally swimming cells (α = 90°) increases with Reynolds number (see Fig. 2a). We should note that a regular array refers to a single squirmer in a periodic computational domain. Thus, the swimming orientation remains constant except for large enough Reynolds number (Re ≥ 10) where inertial effects lead to instability in the swimming orientation of a puller. For Re < 1, this ratio is below 10, which is close to the analytical results in a zero-Reynolds-number regime¹². The distinction between values of both diapycnal eddy diffusivity and Cox number for vertically and horizontally swimming cells significantly increases with Reynolds number (see Fig. 2b,c).

Figure 2

A comparison of mixing induced by vertically moving squirmers in a regular array to horizontally moving squirmers in terms of (a) mixing efficiency, (b) diapycnal eddy diffusivity and (c) Cox number. The corresponding dimensionless parameters are Fr = 5.29, ϕ = 4% and β = 1. All the data points represent the ratio of quasi-steady state values for vertically swimming cells to horizontally swimming cells.

Effect of system size and volume fraction

For a given volume fraction, the effect of the system size has been tested for different number of squirmers, N_s = 1, 8, 12, 27. The temporal evolution of the diapycnal eddy diffusivity K_ρ and Cox number are shown in Fig. 3a,b, where the dimensionless time is calculated as T = tν/a². The results are independent of the system size for N_s ≥ 8³⁷. Unless otherwise stated, we use ϕ = 4% and N_s = 8.

Figure 3

The effect of domain size and initial swimming orientation of squirmers on (a) the diapycnal eddy diffusivity and (b) Cox number is plotted for a swarm of pullers (β = 1) with a volume fraction of ϕ = 4%. The effects of volume fraction of pushers (β = −5) are shown on the (c) mixing efficiency, (d) eddy diffusivity and (e) temperature microstructure. The corresponding dimensionless numbers are Re = 10 and Fr = 5.29. Data points in Figure (c), (d), (e) represent the time-averaged values after the system has reached quasi-steady state; the error bar represents the temporal standard deviation of the time-averaged quantity.

For a given system size, the effect of volume fraction has been evaluated by changing the number of squirmers in the computational domain. The mixing efficiency is almost independent of the volume fraction (Fig. 3c). Both the diapycnal eddy diffusivity and Cox number increase with the volume fraction (Fig. 3d,e).

Effect of density stratification

Density stratification strongly affects the mixing efficiency generated by a suspension of squirmers in a stratified fluid. The mixing efficiency decreases with the Froude number (see Fig. 4a). For Re = 10, the change of the mixing efficiency is about two orders of magnitude from Fr = 3.6 to 52.9. On the other hand, the overall vertical mass flux and temperature microstructures are nearly independent of the density stratification for Fr > 20. The values of diapycnal eddy diffusivity (Fig. 4b) and Cox number (Fig. 4c) slightly increase at small Fr (Fr < 20). It has been noted by Ardekani and Stocker¹³ that the fundamental length scale of the stratification in aquatic environments is . When stratification length scale is larger than the swimmer size, i.e., L/a ~ 1, the behavior is the same as the homogeneous-density fluid and is independent of stratification (L/a ~ 1 corresponds to Fr ~ 30). The local shear generated by swimming ctenophores is about U/a ~ 0.5s⁻¹²³, which corresponds to Fr ~ 50 (N ~ 0.01). Therefore, the vertical mass flux induced by organisms swimming across weak stratification is similar to the one in a homogenous fluid. On the contrary, the fluid stratification strongly reduces the turbulent mixing. The diapycnal eddy diffusivity of the turbulent mixing scales as ε/(νN²) (see black circles in Fig. 1c). The dissipation of kinetic energy in a turbulent flow scales as ε ~ u_t³/l where u_t and l are the characteristic velocity and the integral length scale of the turbulence, respectively. Thus, the turbulent activity parameter can be written as . Consequently, turbulent mixing is suppressed in the region of strong vertical density gradient (e.g. pycnoclines), unlike mixing induced by swimming organisms.

Figure 4

The effect of fluid stratification on the biogenic mixing is quantified.

(a) The mixing efficiency decreases with the Froude number; (b) The normalized diapycnal eddy diffusivity and (c) the Cox number are independent of the Froude number for Fr > 20. The corresponding dimensionless parameters are Re = 10 and ϕ = 4%. All data points represent the time-averaged values after the system has reached quasi-steady state; the error bar represents the temporal standard deviation of the time-averaged quantity.

Effect of buoyancy

Here, we consider the motion of negatively (ρ_p > ρ₀) and positively (ρ_p < ρ₀) buoyant swimming organisms. Figure 5 shows the role of buoyancy on the mixing efficiency, diapycnal eddy diffusivity and Cox number for Re = 10, Fr = 5.29, ϕ = 4%, β = −5. The squirmer’s buoyancy is characterized by b = (ρ_p − ρ₀)/ρ₀. The values of b = 0.0067 and b = 0.03 are selected based on the reported excess density for copepod relative to the ambient fluid which is in the range of 6.7 kg ⋅ m⁻³ to 30 kg ⋅ m^−338,39. Mixing parameters are enhanced for both positively and negatively buoyant swimmers compared to a neutrally buoyant case (b = 0).

Figure 5

The effect of buoyancy on the (a) mixing efficiency, (b) diapycnal eddy diffusivity and (c) Cox number generated by a swarm of pushers (β = −5). The corresponding dimensionless numbers are Re = 10 and Fr = 5.29. All Data points represent the time-averaged values after the system has reached quasi-steady state; the error bar represents the temporal standard deviation of the time-averaged quantity.

Effect of marine turbulence

Turbulent flows are prevalent in marine environments. Therefore, it is necessary to investigate how the organisms interact with the background turbulence⁴⁰. We will show that the biogenic contribution to the total mixing is determined by the magnitude of dissipation of kinetic energy introduced by the organisms. Even though Gregg and Horne’s⁴¹ studies were designed to answer this question, isolation of the effects of the organisms’ swimming from turbulence is difficult, if not impossible, in a natural environment. Direct numerical simulations performed here allow us to separately consider both effects of turbulence and flow disturbances generated by squirmers. We investigate the hydrodynamic interaction of a suspension of swimmers in a decaying stratified turbulence. The initial stratified turbulent flow is generated using direct numerical simulations with a direct forcing approach^42,43, the details of which have been elaborated in the ‘Methods’ section. The Taylor Reynolds number for the turbulence is Re_λ = u_rmsλ/ν ~ 49 and ε/(νN²) ~ 126, where u_rms is the root-mean-square of the turbulent velocity and λ is the Taylor length scale. The corresponding mixing parameters are calculated as K_ρ/ν = 20.56 ± 4.82, COX = 149.28 ± 29.87 and Γ = 0.139 ± 0.027. The value of diapycnal eddy diffusivity agrees well with the curve suggested by Shih et al.³⁴ (see hollow orange square in Fig. 1c). Gregg and Horne took modular microstructure profilers (MMPs) for measuring temperature and velocity microstructure within and outside aggregates of organisms⁴¹ and relevant mixing properties of our turbulence simulation are close to one of their MMPs (MMP15041) outside the aggregate (K_ρ = 1.71 × 10⁻⁵ m²/s, COX = 120, Γ = 0.101). The squirmers’ size is about a ~ 1.07λ and a ~ 14.69η, where η is the kolmogorov length scale. The squirmer Reynolds number is Re = Ua/ν = 100, representing swimming of marine organisms of centimeter size and |β| = 3.

To quantify the relative extent of the biogenic mixing (in the absence of background flow) with regards to the turbulent mixing, a “biomixing active parameter” is defined as

where subscripts b and turb refer to biogenic mixing and turbulent mixing, respectively. The diapycnal eddy diffusivity can be expressed in terms of mixing efficiency Γ and equation (12) simplifies as

For biomixing, Γ_b ≪ 1^9,41 and equation (15) is simplified as K_ρ = Γ_bε_b/(ρ₀N²). For general turbulent mixing in marine environments, Γ_turb ~ 0.17 and K_ρ ~ 0.2ε_turb/(ρ₀N²)³³. Therefore, biomixing active parameter can be expressed as

In our simulations, the temporal evolution of biomixing active parameter B is calculated for a suspension of squirmers and its maximum value is about 0.9 (see Fig. 6d).

Figure 6

A decaying stratified turbulence is modulated by squirmers of Taylor length-scale size.

The transient behavior of the overall mixing is described by the temporal evolution of (a) the normalized diapycnal eddy diffusivity, (b) Cox number and (c) normalized dissipation of kinetic energy. (d) A dynamic evolution of B is shown for both pullers and pushers. (e) A snapshot of a suspension of 8 pushers is shown at T_e = 0.2 (supplementary movie 1 shows swimming dynamics of pushers). (f,g) correspond to trajectories of pullers and pushers, respectively, recorded over 4T_e where different colors distinguish individual squirmers (supplementary movie 2 and 3 show swimming trajectory of pushers and pullers, respectively). The corresponding dimensionless numbers are |β| = 3 and ϕ = 2%. ‘DT’ corresponds to flow with decaying turbulence; ‘DT + Puller’ and ‘DT + Pusher’ correspond to a suspension of pullers and pushers swimming in a decaying turbulence, respectively; ‘Puller’ and ‘Pusher’ correspond to a suspension of pullers and pushers in the absence of turbulence.

The results reveal that pushers more strongly affect the overall mixing compared to pullers. For a decaying stratified turbulence without squirmers (black solid line in Fig. 6a), normalized K_ρ/ν is suppressed within two eddy turnover times, where T_e = t/τ₀ is the normalized time scale and τ₀ is the initial eddy turnover time of the turbulence. With the entrainment of the squirmers of Taylor length-scale size, the dynamics of mixing can be characterized by two distinct stages. In stage I, the value of B for both pushers and pullers is above 0.2 (see Fig. 6d), due to the hydrodynamic interactions between the turbulence and the squirmers’ disturbances. In stage II, the turbulent structure decays away and the flow that is induced by the squirmers is strong. During stage I when T_e < 2, pushers (blue solid line in Fig. 6a) significantly affect the mixing, while the pullers (red solid line in Fig. 6a) do not affect the mixing generated by the decaying turbulence. This is consistent with the results of squirmers in a quiescent flow where pushers (red dotted line in Fig. 6a) are able to generate larger mixing in a stratified fluid than pullers (blue dotted line in Fig. 6a). The dissipation of kinetic energy is also larger for pushers compared to pullers, where ε/ε₀ in Fig. 6c is the normalized dissipation of kinetic energy and ε₀ is the initial dissipation of the decaying turbulence. Similarly, pushers (blue solid line in Fig. 6b) strengthen temperature microstructures in a stratified decaying turbulence, as shown by the dramatic increase of the Cox number compared to the case corresponding to the decaying turbulence in the absence of swimmers (black solid line in Fig. 6b). During stage II, the fine structure of the turbulence decays away. Both the eddy diffusivity and Cox number decrease to their values for a suspension of squirmers in a quiescent flow.

The strong mixing of fluids generated by pushers compared to pullers can be explained by their swimming trajectories. Pushers (Fig. 6g) rectilinearly swim with infrequent changes in their swimming direction due to the squirmer-squirmer interactions, while pullers (Fig. 6f) swim in helical pathes. Even though the slip velocity is axisymmetric for both pushers and pullers, pullers swim in helical trajectories due to their hydrodynamic interactions as well as inertial effects. Our previous results⁴⁴ show that even a single puller at large enough Reynolds number becomes unstable and will not swim on a straight line due to inertial effects. This is not the case for a pusher in the range of Reynolds number investigated in this work. Our previous results for two squirmers moving toward each other in a relatively large computational domain⁴⁴ shows that pullers move on circular trajectories after the collision, whereas swimming direction of a pusher does not change. Over a duration of 4T_e, pushers are much more dispersed compared to pullers (see Fig. 6f and 6g). Consequently, pushers create larger disturbances in stratified fluids. The extent of biogenic mixing depends on the swimming mode of organisms during their migration.

Conclusions

We have numerically studied the hydrodynamic interactions of swimmers with each other, background stratification as well as turbulence. For swimming at Re < 1, the diapycnal eddy diffusivity is generally below the value of molecular diffusion. Therefore, the biogenic mixing at low inertial regime is negligible. Our results show that the mixing efficiency, diapycnal eddy diffusivity and temperature microstructure increase with inertia. The mixing efficiency induced by vertical swimming can be two orders of magnitude larger than horizontal swimming. Vigorous kinetic dissipation generated by a suspension of swimmers enhances the mass transport as well as the temperature microstructure in the presence of background turbulence. Pushers induce larger mixing compared to pullers, which is due to their rectilinear swimming behaviour as opposed to helical motion of pullers. The values of diapycnal eddy diffusivity and Cox number at Re~O(100) are on the same order as the ones caused by turbulent mixing.

Methods

Modeling squirmers in a quiescent background flow

The simulations are conduced in a fixed frame of reference. Equations (3), (4) and (6) are solved in a cubic box of 2π × 2π × 2π. The boundary conditions for u, p and T′ are periodic in all three directions. Convection and diffusion terms in Equation (4) are discretized using QUICK (Quadratic Upstream Interpolation for Convective Kinetics)⁴⁵ and central difference schemes, respectively. Both convection and diffusion terms in equation (6) are solved using central difference schemes to conserve perturbation temperature T′² in space. The temporal discretization is performed using the first-order forward Euler method. The projection method is utilized to enforce the continuity condition in equation (3). The resultant Poisson equation for the pressure is solved using the Hypre library⁴⁶. The tangential squirming motion v_θ on the surface of the spherical particle is satisfied by using a distributed Lagrange multiplier (DLM) technique as explained in our previous publication^29,47. The DLM technique has been broadly utilized to study a suspension of inert particles³⁷ as well as squirmers^19,29. A short-range repulsive force is used to model squirmer-squirmer collision, the details of which are given in our previous work²⁹.

Modeling background turbulence

Turbulence ubiquitously occurs in marine environments. In order to study the hydrodynamic interactions between the turbulence and the disturbances induced by squirmers, a statistically steady stratified turbulence is numerically generated and treated as the initial background flow. Equations (3), (4) and (6) are solved. Instead of calculating the body force using the DLM technique, however, f is a direct forcing term which is given as . The initial temperature field linearly varies with depth. The initial velocity field is generated using an open source code (NTMIX-3D) provided by Centre de Recherche sur la Combustion Turbulente⁴⁸. The initial velocity profile has the following energy spectrum E(k) suggested by Passot and Pouquet (1987)⁴⁹,

where k is the wave number, k_e represents the most energetic wave number and . The stratified turbulence reaches a statistically stationary state as both and ε reach a quasi-steady value⁴³. Once the statistically stationary state is achieved, the direct forcing is no longer required and the stratified turbulence decays. The kolmogorov scale is well resolved in the simulations and ηk_max = 5.23 > 1, where k_max = πN_g/L, N_g = 256 is the number of grid points and L = 2π is the length of the computational domain.