Influence of the Coriolis force on the formation of a seasonal thermocline

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1 DOI /s x Influence of the Coriolis force on the formation of a seasonal thermocline Gahyun Goh & Yign Noh Received: 25 August 2012 /Accepted: 24 July 2013 # Springer-Verlag Berlin Heidelberg 2013 Abstract Large eddy simulation (LES) reveals that the Coriolis force plays an important role in seasonal thermocline formation. In the high-latitude ocean, a seasonal thermocline is formed at a certain depth, across which the downward transports of heat and momentum are prohibited. On the other hand, in the equatorial ocean, heat and momentum continue to propagate downward to the deeper ocean without forming a well-defined thermocline. Mechanism to clarify the latitudinal difference is suggested. The depth of a seasonal thermocline h is scaled in terms of both the Ekman length scale λ and the Monin Obukhov length scale L,ash 0.5(Lλ) 1/2, which is in contrast to the earlier suggestion as h L. seasonal thermocline within the mixed layer, which results in thedecreaseofmld. As to the depth of a seasonal thermocline h, Kraus and Turner (1967) suggested that it is proportional to the Monin Obukhov length scale L (= u * 3 /Q 0 ), where Q 0 is the surface buoyancy flux and u * is the frictional velocity. The equivalent suggestion was also made by Gill and Turner (1976) and Niiler and Kraus (1977). The vertically integrated TKE budget over the mixed layer is given by 0 ¼ G H D; ð1þ Keyword Seasonal thermocline. Coriolis force. Large eddy simulation. Ocean mixed layer. Upper ocean. Turbulence. Stratification 1 Introduction The mixed layer depth (MLD) in the ocean, which plays a critical role in the climate system by controlling the sea surface temperature (SST), shows a strong seasonal variation. Convection generated by surface cooling increases MLD during winter. During summer, however, the balance between turbulent kinetic energy (TKE) generation by wind stress and its decay by surface heating leads to the formation of a Responsible Editor: Yukio Masumoto This article is part of the Topical Collection on the 4th International Workshop on Modelling the Ocean in Yokohama, Japan May 2012 G. Goh: Y. Noh (*) Department of Atmospheric Sciences/Global Environmental Laboratory, Yonsei University, 50 Yonsei-ro, Seodaemun-gu, Seoul , Korea noh@yonsei.ac.kr in the steady state ( h/ t = 0), where G, H, andd represent the contribution from wind stress, surface buoyancy flux, and dissipation, respectively. If buoyancy is assumed to be constant within the mixed layer, H = Q 0 h/2 is obtained. Furthermore, if it is assumed that both G and D are proportional to the TKEfluxbywindstress,orG D=mu * 3, the relation h=2ml is obtained. A wide range of values were suggested for m (m= ; e.g., Gaspar 1988). Alexander and Kim (1976) found,however,that h=2 ml leads to too large h in the midlatitude, and tried to correct it by introducing additional background dissipation. Meanwhile, the fact that the effect of wind stress is limited to the Ekman length scale λ(= u * /f) suggests the possibility that h is influenced by f as well, where f is the Coriolis parameter. With this in mind, several modifications were suggested to predict the depth of a seasonal thermocline, in which the effect of f is taken into account (Resnyanskiy 1975; Elsberry et al. 1976; Garwood 1977; Wells 1979; Gaspar 1988; Noh and Long 1990). These models predict invariably that h decreases with f; for example, h 1/(1/L+1/λ)(Resnyanskiy1975; Garwood 1977). The effect of the Coriolis force can be more naturally included in the mixed layer models using eddy diffusivity, in

2 which the Ekman spiral is reproduced within the mixed layer (e.g., Mellor and Yamada 1982; Large et al. 1994; Noh and Kim 1999). Nonetheless, the indirect effect of the Coriolis force can occur through its effect on the turbulence structure, such as the length scale of turbulence, which must be parameterized in the model (e.g., Cambon et al. 2004; Yeung and Xu 2004; Thiele and Müller 2009). Extensive works have been made to estimate the distribution of MLD in the global ocean with the seasonal and interannual variation from various sources of observation data and ocean general circulation model (OGCM) results (Kara et al. 2003; de Boyer Montégut et al. 2004;Cartonetal.2008; Noh and Lee 2008; Ohno et al. 2009). Attempts have been also carried out to investigate how the mixed layer deepening during winter is related to various factors, such as surface heat flux and wind stress (Deser et al. 1996; Alexander et al. 2000; Yasuda et al. 2000; Tomita et al. 2002; Kang et al. 2010). However, few systematic studies have been done to clarify how MLD is determined in summer, although the relation between h and L under surface heating is suggested in some reports (Alexander and Kim 1976; Schneider and Müller 1990). Recent progress in large eddy simulation (LES), which provides information of realistic three-dimensional turbulence structure in the upper ocean, allows us to investigate the dynamical process of seasonal thermocline formation directly, without resorting to the ad hoc assumption used in the model in previous works (e.g., Sullivan and McWilliams 2010). Therefore, in the present work, we investigate the dynamical process of seasonal thermocline formation using LES, while focusing on the role of the Coriolis force. Furthermore, we obtain the scaling for the depth of a seasonal thermocline. 2 Simulation The LES model used in the present simulation is similar to those used in Noh et al. (2004, 2006, 2009, 2010, 2011), which have been developed based on the PALM (PArallelized LES Model) (Raasch and Schröter 2001). Langmuir circulation is realized by the Craik Leibovich vortex force (Craik and Leibovich 1976), and wave breaking is represented by stochastic forcing. The wave length and height are fixed as 40 and 0.5 m, respectively, for the Stokes velocity used in the vortex force. Simulations without Langmuir circulation and wave breaking are also carried out, however, in order to investigate their effects. Simulation is carried out to reproduce the formation of a seasonal thermocline from the homogeneous upper ocean under the constant stabilizing buoyancy flux, including diurnal cycle, and wind stress at the seasonal time scale. Integration starts by applying the wind stress to the mixed layer with uniform density without the surface buoyancy flux. Once a quasi-equilibrium state is reached after 8 h, the surface buoyancy flux is set on. The wind stress is given by u * =0.01 ms 1, and the surface buoyancy flux is given by eq 0 ¼ Asinð2πt=T Þ, but by eq 0 ¼ B,ifAsin(2πt/T)< B,whereT is 1 day, and A and B are 400 and 100 Wm 2, respectively. In this case, the daily mean buoyancy flux Q 0 results in 81.3 Wm 2. Here, the values of u * and Q 0 are chosen to be consistent with those in the N. Pacific summer (u * ms 1, Q Wm 1 ; Kalnay et al. 1996). Integration is carried out for 10 days, which is expected to be sufficient to reveal the essential dynamics of seasonal thermocline formation. Simulations are performed with various surface buoyancy fluxes (0.1Q 0,0.5Q 0, Q 0,2Q 0 ) and latitudes (ϕ =0,10,20,40, 55 N). The model domain is 300 m horizontally, and 160 m vertically, except the case ϕ =0 o where it is 300 m vertically. The grid size is 1.25 m in all directions. Recently, Noh et al. (2009) simulated the formation of a diurnal thermocline using LES. The dynamical process of seasonal thermocline formation is different from that of diurnal thermocline formation in two important aspects, however. First, a seasonal thermocline is formed over the longer time scale through the cycle of diurnal heating and nocturnal cooling. Second, the depth of a seasonal thermocline is much deeper than that of a diurnal thermocline, so it is formed below the depth affected by the TKE flux due to wave breaking. 3Results 3.1 Formation of a seasonal thermocline Figure 1 shows the time series of profiles of buoyancy B and three terms in the TKE budget, i.e., dissipation rate ε, decay by buoyancy P b, and shear production P s for the cases of ϕ =0 and 40 N. In both cases the upper ocean passes through the diurnal cycle of stratification by surface heating and convection by surface cooling, while the daily mean heat content increases with time. However, Fig. 1 reveals that the response of the upper ocean is radically different depending on latitude. In the high-latitude ocean (ϕ =40 N) the downward transport of B is limited to a certain depth (h 25 m), and ε decreases with time at this depth. On the other hand, in the equatorial ocean (ϕ =0 ), B is transported to the deeper ocean with time. The continuous downward transport of B in the equatorial ocean was also observed in the LES by Wang et al. (1998), who suggested that equilibrium can be only obtained by including the effects of large-scale flows such as upwelling, zonal pressure gradient and the equatorial undercurrent. Furthermore, the present result is consistent with the observational evidences that ε exists in much deeper depths in the equatorial ocean (Peters et al. 1988; Moum et al. 1989) and that the velocity shear, associated with the Ekman spiral in the near-

3 Fig. 1 Time series of profiles of B, ε, P b,andp s : a ϕ =0, b ϕ =40 N. [Scales are different for z and B between (a) and(b)] inertial frequency range in particular, mainly appears at the base of the well-mixed layer during the daily mean surface heating in the high-latitude ocean (Weller and Plueddemann 1996). Accordingly, much larger SST increase at seasonal timescale appears in the high-latitude ocean than in the equatorial ocean, if surface heating is equivalent. P b and P b also appear at increasing depths with time in the equatorial ocean, but they are limited to z=h in the high-latitude ocean. Meanwhile, the negative P b by nocturnal convection reaches to the similar depths (z 20 m) in both cases, indicating that TKE below this depth in the equatorial ocean is generated exclusively by P s. The latitudinal dependence is clearly identified from the evolution of profiles obtained at the end of nocturnal cooling of each day (Fig. 2). Here, stratification N 2 (= B/ z) and velocity shear of the horizontal velocity S 2 (=( U/ z) 2 +( V/ z) 2 ) are plotted instead of P b and P s. Dotted lines represent the profiles before the onset of surface buoyancy flux (t=0 h).

4 Fig. 2 Evolutions of profiles of B, ε, N 2,andS 2 at the end of nocturnal cooling of the day [Δt=24 h and dotted lines represent the profiles before the onset of surface heating (t=0 h)]: a ϕ =0, b ϕ =40 N [Scales are different for z and N 2 between (a) and(b)]

5 Figure 2b indicates that S 2 of the Ekman spiral in the neutral boundary layer (t=0 h) is much smaller than S 2 at the seasonal thermocline. Here, the e-folding scale of the Ekman spiral 0.25λ, estimated by Coleman et al. (1990) from DNS results, corresponds to 25 m. In the high-latitude ocean, a well-defined thermocline appears at z=h ( 25 m) across which B cannot be transported; but, in the equatorial ocean, B continues to penetrate to deeper depths, although it is relatively uniform above z=50 m.in the high-latitude ocean ε continues to decrease with time at z=h; but, in the equatorial ocean, it converges to a certain value at each depth, while penetrating deeper with time. A low level ε maintained below z=h in the high-latitude ocean may represent the radiation of internal waves. The suppression of turbulence prohibits vertical mixing at z=h in the high-latitude ocean, but the maintenance of TKE at a certain level allows vertical mixing at every depth in the equatorial ocean. Consequently, N 2 (z=h) keeps increasing in the high-latitude ocean, but it approaches a certain value at each depth in the equatorial ocean. Note, however, that S 2 (z=h) decreases with time after initial increase in the highlatitude ocean. On the other hand, the profiles of S 2 and N 2 Fig. 3 TimesseriesofP b, P s, ε, N 2,andS 2 at z= m: a ϕ =0 (P b green, P s red, ε blue), b ϕ =0 (N 2 green, S 2 red), c ϕ =40 N (P b green, P s red, ε blue), d ϕ =40 N (N: green, S 2 red)

6 behave similarly in the equatorial ocean. This will be discussed in the following section. 3.2 Mechanism for the latitudinal dependence With an aim to clarify the effect of the Coriolis force in seasonal thermocline formation, we examine the time series of P b, P s, ε, N 2,andS 2 at z= m, which corresponds to z=h in the case ϕ =40 N(Fig.3). Note that these variables are related by P b =K h N 2 and P s =K m S 2,whereK h and K m are eddy diffusivity and viscosity, and the TKE (=E) equation is given by E/ t=k m S 2 K h N 2 ε, if the contributions from TKE flux and vortex force are neglected. In the equatorial ocean, P b, P s, ε, N 2, and S 2 remain invariant with time except the diurnal cycle (Fig. 3a, b). In the high-latitude ocean, however, P s and ε converge to zero, while passing through the diurnal cycle (Fig. 3c). Because of large N 2 at z=h, P b converge to zero at a slightly deeper depth (z>h), as expected from the evolution of B (Fig. 2b). Meanwhile, it is remarkable to observe that N 2 increases with time, but S 2 decreases (Fig. 3d). Based on these features, the feedback mechanism between stratification (N 2 )andtke(e) is suggested as following. Here, all variables represent the daily mean values. The increase of N 2 generated by surface heating decreases E, K h,andk m, which in turn increases N 2 and S 2 by suppressing the downward buoyancy and momentum flux bw and uw. From the relations and, N 2 and S 2 are estimated as proportional to K h 1 and K m 2,ifbw and uw are assumed to be invariant. Therefore, the increase of S 2 (=( U/ z) 2 )ismuch larger than that of N 2 (= B/ z) under stratification, if K m K h, as identified in Fig. 3b. It leads to the increase of P s (=K m S 2 - K m 1 ) with increasing stratification. The increased P s increases E and K h, and it contributes to decrease N 2.Because of this negative feedback, P b, P s, ε, N 2,andS 2 maintain equilibrium for a given depth, as shown in Fig. 3a, b. On the other hand, in the high-latitude ocean, the downward transport of momentum is limited to the Ekman length scale. In this case, the increase of N 2 makes the Ekman spiral shallower, which is evidenced clearly by the decreasing depth of the maximum S 2 with time in Fig. 2b. As a result, the decreased E, K h,andk m, generated by the increase of N 2 under surface heating, cannot induce the increase of S 2 at z=h, as observed in Fig. 3d. The decrease of K m leads to the further decrease of P s at z=h, thus resulting in the further decrease of E and the further increase of N 2 at z=h. Because of this positive feedback, E decreases and N 2 increases continuously with time at z=h. The contrast between two feedback mechanisms is summarized in Table 1. During the diurnal cycle, the maximum N 2 and S 2 appear at the end of surface heating in the equatorial ocean, but they appear at the end of surface cooling in the high-latitude ocean (Fig. 3b, d). Since the downward transport of heat and Table 1 Feedback mechanism between stratification and turbulence The equatorial ocean (f=0) (N ) (K m, K h ) (S, P s ) (K m, K h ) (N ) Negative feedback maintains N, S, and TKE at a certain level at every depth. Continuous downward transport of heat and momentum The high latitude ocean (f 0) (N ) (K m, K h ) (S, P s ) (K m, K h ) (N ) Positive feedback continues to decrease TKE and increase N at a certain depth. Formation of a seasonal thermocline momentum cannot occur across z=h in the high-latitude ocean, stronger convective mixing within the mixed layer enhances N 2 and S 2 at z=h. On the other hand, in the equatorial ocean, higher values of N 2 and S 2 appear, when turbulence is suppressed under surface heating. 3.3 Depth of a seasonal thermocline The previous section clearly showed that h is influenced by f; that is, h=ϕ 1 (u,q 0,f). Dimensional analysis leads to. h ¼ Lϕ 2 λ L ð2þ where λ/l=q 0 /(u * 2 f). Furthermore, h is expected to increase with λ since the vertical extent of momentum transport, triggering the positive feedback mechanism shown in Table 1, is limited by λ. Values of h are evaluated at the end of various experiments with different Q 0 and f, using the definition of the depth with the maximum N 2. Figure 4a reveals an appropriate scaling for h as, which can be replotted equivalently as h 0:5ðLλÞ 1=2 ¼ 0:5u 2 * = ð fq 0Þ 1=2 ð3þ It is in contrast with the earlier suggestion as (Kraus and Turner 1967; Fig.4b). The scaling (3) can be also applied to the equator; i.e., h= if h is defined as the depth until which the downward heat transport reaches (see Figs. 1 and 2). This result is consistent with the analysis that h L leads to too deep MLD (Alexander and Kim 1976). Although the present result confirms the previous works predicting the decrease of h with f (Resnyanskiy 1975; Elsberry et al. 1976; Garwood 1977; Wells 1979; Gaspar 1988), it does not follow the scaling h 1/(1/L+1/λ), suggested by Resnyanskiy (1975) and Garwood (1977). Note that h goes to infinite, as f goes to zero (λ= ), contrary to the prediction from h 1/(1/ L+1/λ). On the other hand, ε persists indefinitely in the

7 Fig. 4 a Variation of h/l with L/ λ. b Variation of h with (Lλ) 1/2 (violet: 0.1Q 0, red: 0.5Q 0, green: Q 0, blue: 2Q 0 ; triangle 10 N, diamond 20 N, circle 40 N, square 55 N) vertical direction, albeit decreasing with z/λ, in the absence of surface buoyancy flux (L= ). It implies that downward transport continues indefinitely (h= ), again contrary to the prediction from h 1/(1/L+1/λ). It is expected that the contribution of wind stress decreases in the presence of the Coriolis force owing to the generation of inertial oscillation, which attributes to limit the downward transport of momentum (e.g., Noh et al. 2010). Furthermore, the vertical component of turbulent flows may be suppressed under the influence of the Coriolis force, in association of twodimensionalization of turbulence (e.g., Noh and Long 1990). As a result of these mechanisms, we can assume that the net contribution of wind stress decreases with the Rossby number λ/h(=u * /fh)asg D u * 3 (λ/h) α,whichleadstoh/l (λ/l) α/(1+α). If α=1, the relation h (Lλ) 1/2 is obtained from (1). The scale (Lλ) 1/2 is also used by Nieuwstadt (1984) and Zilitinkevich et al. (2002) to estimate the equilibrium depth of the stable boundary layer in the atmosphere. The depth of a seasonal thermocline can be estimated by the MLD in summer from observation data. In this case, MLD is usually evaluated by the definition of a certain temperature or density difference from the surface, however, instead of the depth with the maximum N 2, because of practical difficulty. In order to help the comparison with observation data in future works, we reevaluate the depth of a thermocline h * using the definition of ΔB(=B(z=0) B(z=h))= ms 2,whichis roughly equivalent to ΔT=0.2 C. Figure 5 shows that there is no significant difference between the variations of h and h *, although h * does not collapse as well as h. Meanwhile,itis observed that h * tends to be larger than h for smaller f and Q 0, reflecting the fact that the increase of B(z=0)becomes smaller and the thermocline becomes thicker. It makes h * increase slightly faster than (Lλ) 1/ Effects of Langmuir circulation and wave breaking It has been found that both Langmuir circulation (LC) and wave breaking (WB) play a critical role in the formation of a diurnal thermocline by inducing a strong mixing near the sea surface (Noh et al. 2009). On the other hand, these effects can be less important during seasonal thermocline formation because the strong mixing near the sea surface is provided by convection at night even in the absence of LC and WB. Furthermore, the depth of a seasonal thermocline is usually much deeper, where the influences of LC and WB are much weaker. Strong mixing by WB is significant only near the surface. Although LC enhances mixing over the whole mixed layer, its contribution becomes weaker Fig. 5 a Variation of h * /L with L/ λ, b Variation of h * with (Lλ) 1/2 (red:0.5q 0, green: Q 0, blue:2q 0 ; triangle 10 N, diamond 20 N, circle 40 N, square 55 N). (Here, some data in Fig. 4 are missing because ΔB is smaller than ms 2 throughout the depth)

8 Fig. 6 Profiles of B, ε, N 2,andS 2 from the present simulations (solid lines) and from the simulations without LC and WB (dashed lines): a ϕ =0, b ϕ =40 N with increasing depth and stratification (Noh et al. 2011; Sullivan and McWilliams 2010). Nevertheless, it is important to examine how the seasonal thermocline is influenced by these effects.

9 Figure 6 compares the profiles of B, ε, N 2,andS 2 from the present simulation and those without LC and WB at t=6 days. The main feature of seasonal thermocline formation and its dependence on latitude remain essentially invariant, although the thermocline becomes slightly shallower and thicker in the absence of LC and WB in the high-latitude ocean. It is also found that ε tends to be smaller in the absence of LC and WB. 4 Conclusion and discussion In the present work, we have shown that the Coriolis force plays an important role in seasonal thermocline formation. In the high-latitude ocean, a seasonal thermocline is formed at a certain depth, to which the downward transports of heat and momentum are limited. On the other hand, in the equatorial ocean, heat and momentum continue to propagate downward to the deeper ocean without forming a well-defined thermocline. This radical difference is attributed to the fact that there exists a negative feedback between turbulence and stratification in the equatorial ocean, but a positive feedback in the high-latitude ocean. It is also found that the formation of a seasonal thermocline is not significantly affected by wave breaking and Langmuir circulation, although the thermocline becomes slightly shallower and thicker in the high-latitude in the absence of these effects. As a result, the depth of a seasonal thermocline h is scaled in terms of both the Ekman length scale λ and the Monin Obukhov length scale L, and obtained as h=0.5(lλ) 1/2.Itisin contrast to the earlier suggestion as h L (Kraus and Turner 1967). Meanwhile, the present result explains why the scaling h L leads to too deep MLD (Alexander and Kim 1976), and confirms the hypothesis used in several previous mixed layer models that h should decrease with f (Resnyanskiy 1975; Elsberry et al. 1976; Garwood 1977; Wells 1979; Gaspar 1988). The present work predicts the relation h=0.5(lλ) 1/2 = 0.5u * 2 /(fq 0 ) 1/2, which will provide a theoretical basis to estimate the depth of a seasonal thermocline under the ideal condition. However, in the real ocean, h can be affected by various other factors as well. The horizontal heat transport, following the Ekman and geostrophic velocities, can contribute to the heat budget of the mixed layer as well as the surface heat flux. Moreover, downwelling and upwelling induced by the wind stress curl can also modify h significantly. In the equatorial ocean, the presence of a strong subsurface current and the lack of a cooling season make the situation further complicated. It should be also mentioned that the MLD in the real ocean is usually calculated by the criterion of density difference from the sea surface, because of practical difficulty, instead of the criterion of the maximum buoyancy gradient. 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