Contribution of sub-mesoscales to the vertical velocity: The ω-equation

Contribution of sub-mesoscales to the vertical velocity: The ω-equation

Ocean Modelling 115 (2017) 70–76 Contents lists available at ScienceDirect Ocean Modelling journal homepage: Contrib...

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Ocean Modelling 115 (2017) 70–76

Contents lists available at ScienceDirect

Ocean Modelling journal homepage:

Contribution of sub-mesoscales to the vertical velocity: The ω-equation V.M. Canuto a,b,∗, Y. Cheng a,c a

NASA, Goddard Institute for Space Studies, 2880 Broadway, New York, NY, 10025, USA Department Applied of Physics and Mathematics, Columbia University, New York, NY, 10027, USA c Center for Climate Systems Research, Columbia University, New York, NY, 10025, USA b

a r t i c l e

i n f o

Article history: Received 10 June 2016 Revised 11 May 2017 Accepted 13 May 2017 Available online 15 May 2017

a b s t r a c t The ocean’s ability to regulate carbon dioxide depends on biogeochemical processes that are influenced strongly by eddies. Eddy-resolving simulations have shown that sub-mesoscales (SM) (1–10 km) generate the highest magnitude vertical velocity and that mesoscales (M) also enhance their contribution to the vertical velocity but to a lesser extent. In this study, we consider the question: can analogous results be obtained using the less numerically demanding ω-equation? Previously, this question has not been answered because of two reasons: 1) the canonical Hoskins’ form of the ω-equation does not include the buoyancy vertical fluxes caused by M and SM; and 2) Giordani et al. (2016) showed how to include an arbitrary vertical buoyancy flux, but no parameterizations were available for the M and SM vertical fluxes. However, the latter are now available together with their assessments, so we consider SM because they make the largest contribution to the vertical velocity. The resulting vertical velocity depends on the extent of the SM regime, the horizontal buoyancy gradient (representing baroclinic instabilities), and the SM eddy kinetic energy. The vertical velocity depends in a linear manner on the wind stress and it may exhibit seasonal variations. The wind stress has two effects on the ω-equation: indirectly via its contribution to the sub-mesoscale buoyancy flux and directly through the wind stress itself. The results of our sensitivity analysis highlight the range of SM-induced vertical velocities obtained using different input data. © 2017 Published by Elsevier Ltd.

1. Introduction The ocean’s ability to regulate carbon dioxide depends on primary production (PP; Moore and Abbott, 20 0 0), which represents the amount of organic matter attributable to photosynthesis. However, the measured PP is considerably larger than that obtained from traditional nutrient supply models (McGillicuddy et al., 2003), so McGillicuddy et al. (1998, 2007) suggested that eddies may play a significant role in the transport of nutrients to the euphotic zone. High resolution numerical simulations have confirmed this hypothesis and shown that sub-mesoscales (SM) yield the largest vertical velocities (Mahadevan and Tandon, 2006; Legal et al., 2007; Mahadevan et al., 2012; Levy et al., 2001, 2012; Rosso et al., 2014). High resolution codes are highly computationally intensive so in this study, we consider whether the ω-equation can obtain the same results. The canonical form of the ω-equation for the vertical velocity w is:

Corresponding author. E-mail address: [email protected] (V.M. Canuto). 1463-5003/© 2017 Published by Elsevier Ltd.


  ∂ 2w + ∇H · N2 ∇H w = ∇ · Q, 2 ∂z


where f is the Coriolis parameter and N is the Brunt–Vaisala frequency. The first expression for the source term on the righthand side of Eq. (1.1) was derived by Hoskins (1974) and denoted by 2∇ ·Qtg , and it accounts only for geostrophic motion (e.g., Eq. (5) given by Koszalka et al., 2009). Eq. (1.1) has been studied extensively (Pollard and Regier, 1992; Pinot et al., 1996; Capet et al., 2008; Koszalka et al., 2009; Mensa et al., 2013). The limitations of Eq. (1.1) with Hoskins’ form of Q were discussed by Mahadevan (2016). Giordani et al., (2006) derived a “generalized Q-vector” comprising the sum of six terms, where the major difference is that some of them require parameterizations, whereas others do not. Among the latter, the sum of two yields the Hoskins term, and a second term (denoted as the thermal wind imbalance; TWI) has two parts, where one represents stretching and reorientation by the horizontal current fields, as given by Eq. (10) in Giordani et al. (2006), and the other term given by Eq.(11) represents the material derivative of TWI. The remaining two terms are new and they require parameterization, and they are considered in this study. Giordani et al. (2006) called them turbulent forcing

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where they comprise the vertical buoyancy fluxes, which in principle are due to small scale mixing, SM, mesoscales (M), and wind stress. Rosso et al. (2014) showed that a fourfold increase in the resolution corresponds to an approximately twofold increase in the M vertical velocity but to an approximately tenfold increase in the SM vertical velocity, so we focus on SM, the vertical buoyancy flux of which was derived by Canuto and Dubovikov (2010). Using the latter, in principle, we could numerically solve Eqs. (5)–(6) given by Giordani et al. (2006). However, this might be premature because we have not yet estimated the vertical velocity due to the SM buoyancy flux. Thus, we first quantify the effects of SM and wind stresses alone. In particular, we isolate the contributions of these two processes and solve the ω-equation analytically. The vertical velocity is expressed in terms of the horizontal buoyancy gradient, wind stress, depth of the SM regime, horizontal extent, and SM kinetic energy. The wind stress has two effects on the ω-equation: indirectly via its contribution to the sub-mesoscale buoyancy flux and directly through the wind stress itself. 2. ω-equation with buoyancy fluxes and wind stresses Let us consider the following form of the ω-equation given by Eq. (12) in Giordani et al. (2006):

    2 ∂ ∂ 2 τyz /ρ ∂ 2 ∂ Fv ( b ) 2∂ w 2 f +∇H · (N ∇H w )=|f| + , ∂x ∂z ∂ z2 ∂ z2 ∂ x2 (2.1) where the source term on the right-hand side is now represented by the wind stress τ yz and the SM vertical buoyancy flux Fv (b ). To obtain an analytic solution, we note that the ratio of the first term relative to the second term on the left-hand side of (2.1) is (fh/NL)2 > >1, where h is the characteristic vertical scale (tens of meters) and L is the extent of the horizontal region (tens to hundreds of kilometers). Thus, we can neglect the second term, which allows us to solve (2.1) analytically and derive the vertical velocity. As given by Giordani et al. (2006), we adopt the following x, z variations of τyz ,Fv (b ), and h:

τyz (x, z ) = τy (z ) sin θ1 , h(x ) = h0 +h sin θ2 ,

Fv (b )= Fv (z ) sin θ2 , (2.2)


θ1 = κ1 x, θ2 = κ2 (x − xmin ), κ1 = π /xmax , κ2 = π /(xmax − xmin ),


where xmin , xmax denote the horizontal extent of the region under consideration (see Section 7 for the values used in this study). Then, Eq. (2.1) has the following form:


∂ 2 τy ( z ) ∂ 2w ∂ Fv ( b ) = κ1 |f| cosθ1 −κ 22 sinθ2 , 2 ∂z ∂z ∂ z2



and thus (3.1) can be expressed in a manner that highlights the contributions of wind and SM:

w(x, z ) = wwind (x, z )+wSM (x, z ) wwind (x, z ) = wwind (z )cosθ1 , wSM (x, z ) =wSM (z )sinθ2 ,


where the z-dependences of the wind and SM contributions are given as follows. −1

W ind : wwind (z ) = κ1 |f|

T ( z ), z T(z ) ≡ τy (z )−τy (0 )+ [τy (−h )−τy (0 )] h

SM :


z wSM (z ) = − κ22 f−2 [I(z )+ I(−h )] h


The physical variables characterizing these problems are:

h, f , ∇H b, Ro, EBF,



where ∇H b is the horizontal buoyancy gradient (baroclinic insta/2 bilities), b = − gρ −1 ρ is the buoyancy, Ro ∝ K1SM /(rs N ) is the SM 0 Rossby number, KSM is the SM kinetic energy, rs = O(1km) is the horizontal extent of SM, EBF = f−1 ρ −1 τw ×ez · ∇H b is the Ekman buoyancy flux, and τ w is the surface wind stress. 4. Contribution of wind stress to w To obtain the actual results, we consider the following two relations.

fez × uag = ρ −1 ∂z τ (z ),

ρνt ∂z uag = τ (z )


The first relation is the definition of the wind component of the two-dimensional (2D) mean velocity, which we denote as ageostrophic, and the second is a “closure” model, where ν t is the vertical momentum diffusivity due to small scale mixing. If we combine the two relations (4.1), the result is a differential equation for τ (z). If we assume that the vertical momentum diffusivity is constant in the mixed layer, then the equation has the following solution, which can be verified directly:

τ (ζ ) = α (ζ )τw +f|f|−1 β (ζ )ez ×τw , α (ζ ), β (ζ ) ≡ eζ (cosζ , sinζ ), ζ = z/δE (4.2) where δ E = (2ν t |f| − 1 )1/2 is the Ekman depth. α (0) = 1, β (0) = 0, and τ w represents the surface wind stress τ (0), so it is interesting to determine the averages over the SM extent h defined 0 as < A > = h−1 −h A(z )dz : 2 < α (ζ ) >=E1/2 , 2 < β (ζ ) >= −E1/2 , where E =(δ E /h)2 is the Ekman number. By substituting (4.2) into (3.4), we obtain: −1

wwind (z ) = κ1 |f|

T ( z ),

T0 ( z ) = α ( ζ ) − 1 +

T(z ) =T0 (z )τwy + T1 (z )

z (α∗ − 1 ), h

T1 ( z ) = β ( ζ ) +



(ez × τw )y

z β∗ , h


where α ∗ ≡ α (-E-1/2 ), β ∗ ≡ β (-E-1/2 ). Thus, by substituting (4.3) into the first part of (3.4), the contribution of the wind stress wwind (z) to the vertical velocity is determined.

which we solve in the following sections. 3. Boundary conditions

5. SM vertical buoyancy flux The general solution of (2.4) is given as follows.

f2 w = κ1 |f|τy (z ) cos θ1 − κ22 I(z ) sin θ2 + c1 z+c2 ,  z I (z ) ≡ Fv (z )dz



Imposing the boundary conditions that w = 0 at z = 0, -h, leads to:

c2 = −κ1 |f|τy (0 ) cos θ1 ,

hc1 = κ1 |f|[τy (−h ) − τy (0 )] cos θ1 − κ22 I(−h ) sin θ2 ,


In principle, the availability of parameterizations for the vertical buoyancy fluxes due to M and SM allows us to include both of them in the last term in (3.1). We could even include the vertical buoyancy flux due to small scale mixing, while M would restratify the flow (Oschlies, 2002), whereas small scale mixing does the opposite and de-stratifies it. In the present study, we limit ourselves to considering the effect of SM because, as mentioned in Section 1, numerical simulations have shown that they contribute the most to increases in the vertical velocity. To introduce


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the SM model, we first consider the case of a geostrophic flow where the variables that characterize the system are h (depth of the SM regime), f (Coriolis parameter), and ∇H b (horizontal buoyancy gradient), which yields the following buoyancy flux by using dimensional arguments: Fv (b )  h2 |f|−1 |∇H b|2 . The dependence on h2 may imply that during the winter when the mixed layer is deep, this formula could overestimate the flux, whereas the flux may be underestimated in the summer when the mixed layer is shallow. The expression presented below comprises both the geostrophic and ageostrophic components of the velocity field, and it has been shown to satisfactorily reproduce both the summer and winter data for the buoyancy flux. a) General form As derived by Canuto and Dubovikov (2010), the SM buoyancy vertical flux has the following form:

Fv (b ) = − A(Ro )κˆ · ∇H b A(Ro ) ≡ Ro (1 + Ro ) 2

2 −1

1 f κˆ ≡ κ− e z ×κ , Ro |f|

, ,


where the diffusivity κ and Ro are defined as follows.

κ (z ) ≡ Ro =

z 0

[u(z )− < u>]dz

(2KSM )1/2 , rs |f|

rs =


π |f|

(5.2) = O(1km )


The parameterization requires a model for KSM , which we address in Appendix A. To give the reader an idea of the value of Ro, we use the numerical simulation of the California Current given by Capet et al. (2008), which provides the following values, KSM ࣃ 10 − 3 m2 s − 2 , N = 10 − 3 s − 1 and h = 40 m; thus, relations (5.3) then give Ro = 3.5. Some comments on relations (5.1)–(5.3) may be useful. The presence of the 2D mean velocity u(z ) in the parameterization of the buoyancy vertical flux may seem unusual but its origin in straightforward. The buoyancy flux is defined as Fv (b ) = w b where the primes indicate SM. We begin by considering the dynamic equations for the total buoyancy and velocity fields, where the latter are separated into the mean and fluctuating parts. After inserting them into the original equations and averaging, we obtain the dynamic equation for the fields w , b . After multiplying the first by b’, and the second by w’, adding the results, and averaging, we have the dynamic equation for the correlation w b . The advective term in the starting equations contains the 2D mean velocity u(z ), which then becomes part of the vertical tracer flux (5.1). Furthermore, we may question why the SM flux involves the SM kinetic energy, whereas previous parameterizations did not, but its presence is expected on physical grounds because as KSM goes to zero, this means that no sources can sustain the SM and thus there cannot be a buoyancy flux. In fact, we can observe that the buoyancy flux (5.1) vanishes in the limit Ro = 0. Its origin is attributable to the nonlinear nature of the problem. In fact, if we use a linear model, a velocity second-order correlation such as KSM cannot be computed because linear models cannot determine the amplitudes. By contrast, in the nonlinear model, the original SM

dynamic equations contain a turbulent viscosity/diffusivity.1 Using the mixing length model, the latter are the product of the velocity times the length, and in (5.3), the former appears as the square root of the eddy kinetic energy KSM and the latter as rs representing the SM horizontal extent. Nonlinearity also explains why the diffusivity in (5.2) contains an integral, i.e., because turbulent mixing is intrinsically nonlocal and at any z, the diffusivity is contributed by all values of z from z = 0 to z. b) Geostrophic and ageostrophic contributions to the SM buoyancy flux The 2D mean velocity u(z ) can be obtained from an OGCM (Oceanic General Circulation Model), but we consider the following form to obtain analytic solutions of the new ω-equation:

u(z ) =ug + uag , ug = f−1 zez × ∇H b


where in the geostrophic component, we have used the fact that the horizontal buoyancy gradient is constant in the mixed layer (Mensa et al., 2013; Veneziani et al., 2014). The wind (ageostrophic) component was defined in (4.1). By inserting (5.4) into (5.2) and then into (5.1), the two components of the vertical buoyancy flux are derived as (we consider only down-front winds):

Fv (b ) = Fv (b ) + Fag v (b ) g

Fgv (b ) = −


2 1 A(Ro ) h2  ∇H b σ (1 + σ ), 2 Ro |f|

Fag v (b )= A (Ro )λ (ζ )EBF, (5.6)

where EBF was given after Eq. (3.6) and we have introduced the two dimensionless variables: σ ≡ z/h and λ(ζ ) = 1 + σ − α (ζ ) − Ro − 1 β (ζ ). At the surface, z = 0, and both fluxes (5.6) vanish. We also note that the geostrophic component recovers the form discussed in the first part of this section.

6. Contribution of SM to w By substituting (5.5)–(5.6) into (3.5), the geostrophic and ageostrophic contributions to wSM become:

wSM (z ) =wg Ig (σ ) + wag Iag (σ ) wg =

2 A(Ro ) 3 −3 2  h |f| κ2 ∇H b , 4Ro

wag = −A(Ro )κ 2 f−2 hEBF 2

where thedimensionlesfunctionsIg (σ ),Iag (σ ) are given as follows.


2 2 σ , Iag (σ ) = I(σ ) + σ I(−1 ) 3 3

1 1 1/2 1 I ( σ ) = σ 1 + σ − E1/2 ( α + β − 1 ) − E (β − α + 1 ) 2 2 2Ro (6.2) Ig ( σ ) ≡ σ

+σ +

Table 1 Maximum values of the vertical velocities for δ E = 10 m. Case

− by (s − 2 )

τ w (Nm − 2 )

wwind (x,z) (m/day)

wSM,g (x,z) (m/day)

wSM,ag (x,z) (m/day)

w(x,z) (m/day)

1 2 3 4 5 6

10 − 7

0.2 1.0 0.2 1.0 0.2 1.0

0.73 3.66 0.73 3.66 0.73 3.66

0.012 0.007 0.50 0.43 16.2 15.5

0.05 0.27 0.55 2.73 5.46 27.3

0.73 3.66 0.92 3.69 20.2 37.4

10 − 6 10 − 5


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Table 2 Maximum values of the vertical velocities for δ E = 30 m. Case

− by (s − 2 )

τ w (Nm − 2 )

wwind (x,z) (m/day)

wSM,g (x,z) (m/day)

wSM,ag (x,z) (m/day)

w(x,z) (m/day)

1 2 3 4 5 6

10 − 7

0.2 1.0 0.2 1.0 0.2 1.0

0.72 3.61 0.72 3.61 0.72 3.61

0.013 0.009 0.50 0.44 16.3 15.7

0.04 0.20 0.39 1.95 3.81 19.0

0.72 3.60 0.82 3.43 19.4 32.4

10 − 6 10 − 5

Fig. 1. Upper panel: contour map of the SM-induced vertical velocity geostrophic component (m/day) wSM,g (x,z), see Eq. (6.3). Lower panel: the ageostrophic component (m/day) wSM,ag (x,z), see Eq. (6.3). Ro is determined from Eq. (A.2). The data used correspond to Case 6 in Table 1.


V.M. Canuto, Y. Cheng / Ocean Modelling 115 (2017) 70–76

Fig. 2. Upper panel: contour map of the wind-induced vertical velocity (m/day) wwind (x,z), Eq. (4.3). Lower panel: total vertical velocity (m/day) w(x,z), Eq. (3.3). Ro is determined from Eq. (A.2). The data used correspond to Case 6 in Table 1.

The relations (6.1) require some comments. Using (6.1), wSM (x,z) in (3.3) can be split into the following two components.

wSM (x, z ) = wSM,g (x, z )+wSM,ag (x, z ) wSM,g (x, z ) = wg Ig (σ )sinθ2 ,

wSM,ag (x, z ) =wag Ig (σ )sinθ2 (6.3)

The horizontal buoyancy gradient ∇H b acts as a source of SM, so we expect that the vertical velocity will increase as ∇H b increases. In particular, wg scales as |∇H b|2 , the ageostrophic com1 A simplified description of the treatment of nonlinearity can be found at: http://, as well as assessments in different turbulent flows and the relevant references.

ponent wag depends on EBF and is linear in ∇H b, and for a fixed wind stress, the increase with ∇H b is less than that in wg . However, the complete dependence of the two velocities on ∇H b must consider that Ro also increases with ∇H b and the final dependence of wg on ∇H b is less than quadratic. By contrast, wag does not depend on Ro and its increase with ∇H b is linear. The dependence on h is quite different for wg and wag . h is deep in the winter and shallow in the summer, so the two contributions differ in the winter and summer. In the winter, the geostrophic component is likely to be larger than the ageostrophic component, whereas the opposite occurs in the summer. However, if the wind stresses are stronger in the winter than the summer,

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the weaker dependence of wag on h may be somewhat compensated. The dependence of the Coriolis parameter f is the same for both wg and wag because EBF contains an f − 1 factor. The pure wind contribution depends only on the wind stresses that enter in a linear manner in the vertical velocity, and thus there is only a possible seasonal dependence. The relations (6.1) exhibit a strong dependence on the components that are location dependent, so it is not possible to define a characteristic vertical velocity. The best way to illustrate these results is by performing a sensitivity analysis, as shown in Section 7. The following two limits are also of interest. a) Low SM kinetic energy, Ro < 1:

 2 1 −3 wg = Roh3 |f| κ22 ∇H b , 4

wag = −Ro



2 −2 2 f hEBF


As expected, a low kinetic energy SM yields low vertical velocities, which are proportional to Ro and Ro2 . b) Large SM kinetic energy, Ro > 1:

wg =

2 1 3 −3 2  h |f| κ2 ∇H b , 4Ro

wag = − κ22 f−2 hEBF


7. Sensitivity tests The values for the variables in (2.2)–(2.3) are the same as those used by Giordani et al. (2006):

xmin = 200km, xmax = 300km, h0 = 40m, h = 80m.


In Tables 1 and 2, we present the maximum values of the vertical velocity (m/day) due to the wind, the geostrophic and ageostrophic components (6.1), and the total vertical velocities for Ekman depths of δ E = 10m, 30m. We consider three values for the horizontal buoyancy gradient and two values for the wind stress, where the values of Ro, which depend on by , τ w , and h, are obtained by solving Eq. (A.2). The smaller and larger values for the buoyancy horizontal gradients correspond to the values found in the California Current (Capet et al., 2008) and in the study by D’Asaro et al. (2011). The different vertical velocities reflect the strong dependences on local values. According to the values in the tables, we can observe that the maximum value of wg does not scale exactly with by 2 because when Ro > 1, wg decreases as Ro − 1 slows down the rate of increase with by 2 . In Figs. 1–2, for case 6 in the tables, the maximum value of w(x,z) does not occur at the same (x, z) and the maximum of the total w(x,z) (column 7) is not the sum of the maximum values of its components (columns 4–6). 8. Conclusions The aim of this study was to quantify the contributions of SM buoyancy fluxes and wind stress to the vertical velocity. Using the SM vertical buoyancy flux assessed based on a variety of data, we found that the ω-equation Eq. (2.1) allows an analytic solution given by Eq. (3.3), where the vertical velocities due to SM are given by (6.1)–(6.2) and the contribution due to wind stresses is given by (4.3). It should be noted that the wind stresses have two effects on the ω-equation: indirectly via the effects on the SM buoyancy flux, i.e., the second relation in (6.1), and directly through the stresses themselves, as discussed in Section 4. Tables 1 and 2 present some results to highlight the dependence of the vertical velocity on the variables considered in the problem. The existing numerical codes for the ω-equation include the Hoskins’ term 2∇ ·Qtg , so the next step is to add the SM buoyancy flux and wind stresses. For the former, we must use relations (5.1)– (5.3), where u(z ) can be obtained directly by the OGCM, the SM kinetic energy in terms of large scale variables is given by relation (A.2), and h is determined as discussed by Canuto et al. (2017).


Acknowledgements The authors would like to thank the Editor, Dr. A. Oschlies, and two anonymous referees for several useful recommendations, which helped improve the presentation of this study. Resources for this study were provided by the NASA High-End Computing (HEC) Program through the NASA Center for Climate Simulation (NCCS) at the Goddard Space Flight Center. All the results obtained in this study are available from the authors. Appendix A. SM kinetic energy According to Eq. (5.3), we found that in the California Current, Ro = 3.5. However, we expect Ro to be location dependent, so it is not justifiable to use this value of Ro for other locations. Thus, we require a relation for Ro in terms of the large-scale, locationdependent variables, which comprise the geostrophic Richardson number, EBF, and the wind stresses represented by the dimensionless combinations, as follows.

Rig =

f2 N2

  , ∇H b2


EBF h2 N2



τ∗ ≡

(τw /ρ )2 h4 N2 f2


Some physical considerations are also required. Ro must decrease with Rig ; in fact, when Rig is larger, the SM source represented by baroclinic instabilities (horizontal buoyancy gradient) is weaker and the SM kinetic energy that they generate the must be lower. However, at a fixed Rig , winds act as a source of EKE (Eddy Kinetic Energy) and Ro must increase with τ ∗ . These features are reproduced by the solution of the algebraic equation derived by Canuto et al. (2017): −1

Ro3 = 0 Rig + 1 e+2 τ∗ ,


where the dimensionless  functions are defined as follows:

12q0 π 2 0 = A(1 +2A )Ro−1 , 2q0 π 2 1 = A[1 − E1/2 (1 − Ro−1 )] 1 + A2 Ro−1 , q0 π 2 2 = 2A2 Ro−1 E−1/2 (1 − E1/2 ) (A.3) 3 3/2

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