The KPP boundary layer scheme: revisiting its formulation and benchmarking one-dimensional ocean simulations relative to LES

Transcription

1 The KPP boundary layer scheme: revisiting its formulation and benchmarking one-dimensional ocean simulations relative to LES Luke Van Roekel a, Alistair J. Adcroft b, Gokhan Danabasoglu c, Stephen M. Griffies b, Brian Kauffman c, William Large c, Michael Levy c, Brandon Reichl b, Todd Ringler a, Martin Schmidt d a Fluid Dynamics and Solid Mechanics, Los Alamos National Laboratory, Los Alamos, NM, LA-UR b Geophysical Fluid Dynamics Laboratory, Princeton, NJ c Climate and Global Dynamics Laboratory, National Center for Atmospheric Research, Boulder, CO d Leibniz-Institute of Baltic Sea Research, Warnemünde, Seestraße 15, 18119, Rostock, Germany Abstract We evaluate the Community ocean Vertical Mixing (CVMix) project version of the K-profile parameterization (KPP). For this purpose, onedimensional KPP simulations are compared across a suite of oceanographically relevant regimes against large eddy simulations (LES). The LES is forced with horizontally uniform boundary fluxes and has horizontally uniform initial conditions, allowing its horizontal average to be compared to one-dimensional KPP tests. We find the standard configuration of KPP (Danabasoglu et al., 2006) consistent with LES results across many simulations, supporting the physical basis of KPP. We propose some adaptations of KPP for improved applicability relative to LES comparisons. First, KPP requires that interior diffusivities and gradients be matched to KPP predicted values. We find that difficulties in representation of derivatives under rapidly changing diffusivities near the base of the ocean surface boundary layer (OSBL) can lead to loss of simulation fidelity. We propose two solutions: (1) require linear interpolation to match interior diffusivities and gradients and (2) a computationally simpler alternative where the KPP diffusivity is set to zero at the OSBL base and interior mixing schemes are extended to the surface. Yet the latter approach is sensitive to implementa- Preprint submitted to Ocean Modeling May 10, 2017

2 tion details and we suggest a number of solutions to prevent emergence of numerical high frequency noise. Second, the traditional cubic shape function in the KPP non-local tracer flux can potentially lead to spurious stable stratification near the surface of the boundary layer, e.g., warming the upper ocean under surface cooling. Tests with alternative non-local shape functions are more consistent with LES. Further, the chosen time stepping scheme can also impact model biases associated with the non-local tracer flux parameterization. While we propose solutions to mitigate the artificial near-surface stratification, our results show that the KPP parameterized non-local tracer flux is inconsistent with LES results in one case due to the assumption that it solely redistributes the surface tracer flux. In general, our reevaluation of KPP has led to a series of improved recommendations for use of KPP within ocean circulation models based on a test suite of LES results. The test suite can be used as an oceanic baseline for evaluation of a broad suite of boundary layer models. Keywords: KPP, large eddy simulation, Vertical Mixing, Mixed Layer, Boundary Layer, parameterization Draft from May 10, Introduction The ocean surface boundary layer (OSBL) mediates momentum, heat, and tracer fluxes between the atmosphere and cryosphere with the interior ocean. Consequently, an accurate parameterization of turbulence and the induced vertical mixing in the OSBL is essential for robust and reliable model simulations of climate physics as well as the abundance and distribution of important biological and chemical quantities. We here consider the formulation and behavior of the K-profile parameterization (KPP) (Large et al., 1994, LMD94), which has been used by a variety of ocean and climate applications. Here we consider one-dimensional vertical fluxes only. KPP fidelity in the presence of horizontal features (e.g., fronts; Bachman et al., 2017) is not considered. We test the CVMix implementation of KPP against LES configured with similar depth and surface forcing. We furthermore compare KPP results realized in three ocean circulation models: Model for Prediction Across Scales 2

3 Ocean (MPAS-O; Ringler et al., 2013); Modular Ocean Model Version 6 (MOM6; Adcroft and Hallberg, 2006); and the Parallel Ocean Program Version 2 (POP2; Smith et al., 2010). We hypothesized that no differences would arise between the models, as the same CVMix code is used to implement KPP. However, simulation differences can arise due to details of how KPP is embedded within the model time stepping scheme. We found these details to explain some differences in behavior relative to LMD94 and to be very important for evaluating the integrity of KPP and for choosing its parameters. This paper is organized into two main parts. The first part reviews various aspects of how OSBL schemes, with a focus on the KPP scheme, are formulated physically and numerically. This part consists of Section 2, where we a subset of present OSBL schemes and critical underlying assumptions, Section 3, where we summarize elements of the KPP scheme, and Section 4, where we present salient implementation considerations for KPP along with issues that arose in our development and evaluation. The second part of the paper comprises our experiences benchmarking KPP in comparison to LES. We discuss the test cases and LES model in Section 5, and then present results and analysis in Section 6. For identical configurations of CVMix, MOM6, MPAS-O, and POP produce similar results. Thus sensitivity experiments will consider MPAS-O only. The presentation of results focuses on the issues discussed in Section 4 and possible solutions. In closing, we present conclusions and recommendations for best practices and future KPP developments in Section 7. Various appendices offer further details about aspects of the KPP scheme and the test cases. 2. A Survey of Boundary Layer Mixing parameterizations Generally, boundary layer parameterizations assume the turbulent mixing is dominated by vertical fluxes. Presently, varying degrees of complexity are used to simulate these vertical fluxes. Bulk boundary layer models are perhaps the simplest of these models (e.g., Kraus and Turner, 1967; Niiler and Kraus, 1977; Price et al., 1986; Gaspar, 1988). These models assume that ocean properties (e.g., tracers and momentum) are vertically uniform in the OSBL. The boundary layer depth in this class of models is diagnosed via either; a prognostic equation based on layer integrated turbulence kinetic energy (TKE; Kraus and Turner, 1967; Gaspar, 1988) or a critical Richardson number criterion (Price et al., 1986). The assumption of no vertical structure 3

4 within the boundary layer is both the key simplification and the main deficiency of bulk models. Namely, tracers and velocity are not generally uniform vertically, even in the presence of strong mixing. Hence, vertical integration over the depth of the OSBL precludes the simulation of OSBL processes such as the Ekman spiral and boundary layer restratification (Hallberg, 2003). TKE closure (TC) is another widely used framework for parameterizing upper ocean boundary layer mixing (e.g., Kantha and Clayson, 1994; Canuto et al., 2001; Umlauf and Burchard, 2005; Canuto et al., 2007; Harcourt, 2015). In most variants of TC, the profiles of eddy diffusivity and viscosity are dependent on the local TKE, which is prognostic (e.g., Mellor and Yamada, 1982; Kantha and Clayson, 1994). These models are local, since the turbulent fluxes of tracers and momentum are only proportional to the square root of the local TKE and vertical derivative of the mean field. For example, the turbulent vertical flux of potential temperature in these models takes the form ( ) θ w θ K (1) z where K l e (2) is the vertical eddy diffusivity with l a turbulent length scale and e the TKE 1. In equation (2) it is generally assumed that the dissipation scales as the product of a turbulent length scale and e 3/2 (Batchelor, 1953) to convert from the more traditional dissipation based mixing definition (Osborn, 1980). In highly convective conditions, turbulent fluxes are dominated by processes unrelated to local gradients since strong fluxes exist even when θ/ z 0. Thus traditional TC models cannot correctly simulate highly convective conditions, as they predict a zero flux in much of the boundary layer. Furthermore, typical TC models often predict too little mixing across stable stratification (e.g., Martin, 1985; Kantha and Clayson, 1994; Canuto et al., 2007). When applied to the ocean, the K-profile parameterization (KPP) (Large et al., 1994, LMD94) aims to alleviate the above two limitations of TC methods. It also aims to represent a middle ground between bulk boundary layer 1 More precisely, e is the turbulent kinetic energy per mass, so its dimensions are squared length per squared time. Hence, the diffusivity, K, has dimensions of squared length per time. 4

5 models and prognostic TC models. KPP allows for vertical property variations in the OSBL and includes a parameterized non-local transport. KPP also alleviates the under prediction of mixing seen in TC models in certain forcing scenarios (Canuto et al., 2007). In KPP, the vertical structure of eddy viscosity and diffusivity is specified by a vertical shape function (O Brien, 1970). This function is matched to Monin-Obukhov similarity theory at the surface and matches the interior diffusivity (or transitions to zero) at the OSBL base (see Figure 1). KPP only operates within the OSBL and it has no prognostic equation for TKE or length scale. KPP is thus not subject to energetic constraints present in TC models. Consequently, when the OSBL shallows, The lack of energetics makes it difficult for KPP to capture transitions when the OSBL shallows. For example, the KPP contribution to mixing immediately ceases at depths below a shoaling OSBL, thereby relying on interior parameterizations to represent any remnant mixing. Furthermore, the lack of TKE consideration in prediction the OSBL depth could result in premature shoaling or deepening of the OSBL. As with many boundary layer parameterizations, there is a strong dependence of the diagnosed boundary layer properties on physical and numerical specifics of KPP (e.g., Vogelzang and Holtslag, 1996; Seidel et al., 2010; Lemarié et al., 2012; McGrath-Spangler and Molod, 2014; McGrath-Spangler et al., 2015; McGrath-Spangler, 2016) and interpolation between the OSBL and interior (e.g., Danabasoglu et al., 2006; Seidel et al., 2010). We explore a subset of these dependencies within this paper Evaluations of select OSBL parameterizations We here summarize certain methods used to evaluate OSBL parameterizations. We argue for the use of LES methods to define the baseline solution and mention the value of testing the CVMix implementation of KPP within multiple ocean circulation models Evaluations based on comparisons to observed field data One-dimensional vertical mixing models have been tested against profile data in configurations where horizontal advective tendencies are small (e.g., Price et al., 1986; Large et al., 1994; Kantha and Clayson, 1994; Zedler et al., 2002; Wijesekera et al., 2003; Van Roekel and Maloney, 2012; Calvert and Siddorn, 2013; Mukherjee and Tandon, 2016). Biases relative to observations vary between studies. For example, Van Roekel and Maloney (2012) 5

6 Surface fluxes B R, B f, τ, Q m Air-sea or ice-sea interface: z = η MO surface layer ɛ h Penetrative shortwave KPP boundary layer thickness: h ocean bottom: z = H Figure 1: Schematic of the upper ocean regions associated with the KPP boundary layer parameterization. The upper ocean is exposed to fluxes of momentum, τ, mass, Q m, and buoyancy, B f, at the air-sea interface. Penetrating shortwave radiation, (w θ ) R, along with its associated buoyancy flux, B R > 0, also influence the upper ocean. The Monin- Obukhov (MO) surface layer transfers these fluxes to the remainder of the boundary layer. Here we assume the surface layer extends from the free surface (z = η) to z + η = ɛh, where h > 0 is the depth of the boundary layer base (typically ɛ 0.1). 6

7 and Mukherjee and Tandon (2016) found model biases changed dramatically during the simulation, where KPP tends to over entrain in stable conditions and was otherwise consistent with observations. Zedler et al. (2002) found KPP performed better than a bulk and TC model in hurricane winds, attributed to the inclusion of the LMD94 internal shear instability scheme via diffusivity matching in KPP. OSBL parameterizations have also been tested in three-dimensional configurations (e.g., Li et al., 2001; Durski et al., 2004; Canuto et al., 2004; Chang et al., 2005). However, given the relative paucity of observations and large biases in surface fluxes, it is difficult to draw clear conclusions regarding parameterization performance. We therefore make use of one-dimensional simulations in this study Evaluations based on comparisons to large eddy simulations LES provide our chosen evaluation framework for KPP. In LES, the dominant eddies are explicitly resolved and the sub-grid turbulence is small relative to that at resolved scales. The LES model is forced with horizontally uniform fields and the output is horizontally and temporally averaged, thus allowing us to neglect horizontal processes. We compare the LES results to those from one-dimensional KPP simulations using identical forcing and initialization. Numerous comparisons of atmospheric boundary layer parameterizations relative to LES have been conducted (e.g., Holtslag and Moeng, 1991; Moeng and Sullivan, 1994; Brown, 1996; Ayotte et al., 1996; Noh et al., 2003). Yet there are few ocean LES evaluations. Many of the oceanic comparisons that have occurred make use of similar (and limited) initial conditions and forcing scenarios (e.g., McWilliams and Sullivan, 2000; Smyth et al., 2002; Reichl et al., 2016; Bachman et al., 2017). Following atmospheric comparisons (e.g., Ayotte et al., 1996), we use a suite of forcing and initial conditions in our LES experiments. Unlike previous ocean LES comparisons (e.g., Wang et al., 1998; Large and Gent, 1999), our experiments do not depend on possible biases in observations and ocean model output that inherently depends on the vertical mixing scheme being tested. Thus, our tests seek a clear method to examine KPP implementation choices, and are not as complex as some previous ocean LES comparisons (e.g., Large and Gent, 1999). 7

8 The CVMix version of KPP CVMix is a collection of vertical mixing parameterizations. It is developed as a suite of kernels to be implemented in a three-dimensional ocean circulation model. The CVMix project aims to address the needs of research groups to code, test, tune, and document parameterizations of oceanic vertical mixing for numerical ocean simulations. Notably, CVMix does not determine time stepping for the model prognostic fields. Instead, time stepping is the responsibility of the calling circulation model. As a part of the CVMix development, our purpose here is to evaluate the influence of physical assumptions and numerical choices within the CVMix version of KPP as well as in the ocean circulation model 2. One such example is the change in present calling ocean models from the proposed semi-implicit time stepping scheme from LMD94. In LMD94, the time stepping involves an iteration to convergence of the boundary layer depth diagnosed by the end-state of a time step. Such an iteration is not performed by the OGCM schemes, because of the complexity and computational demands. Instead, the boundary layer depth is diagnosed from earlier time levels. This change may explain some of the issues identified in Section 6.1. KPP has been implemented in numerous ocean circulation models. However, in our experience each implementation makes distinct physical and numerical choices. Sometimes, these implementation choices can have unintended consequences which can negatively impact the KPP boundary layer simulation. As shown in this paper, many choices have nontrivial impacts. This situation provided the mandate for our development of KPP within the CVMix project (Griffies et al., 2015). The present paper serves to document many of our findings Comments on the limitations of OSBL parameterizations OSBL schemes aim to parameterize correlations of vertical velocity and property fluctuations (e.g., see equation (3) below). These correlations are averaged over a fairly broad range of spatial scales (including the grid scale) and over the model time step. As model grid spacing and time step are refined, the associated averaging scales change. We therefore expect that OSBL models will change behavior with refined space-time resolution. Furthermore, the parameterizations may not smoothly transition to a model that 2 For brevity, in this paper KPP refers to its CVMix implementation. 8

9 explicitly resolves small scale turbulent boundary layer processes. Thus, it is possible that the subgrid scale averaging imposes a window of applicability for a given OSBL parameterization based on the model time step and grid spacing. A similar issue with the cloud physics convergence problem is discussed by Jung and Arakawa (2004). OSBL models are forced by surface fluxes that are uncertain and strongly variable on small temporal and spatial scales (e.g., Fairall et al., 2003; Large and Yeager, 2009; Brunke et al., 2011). Further, the surface forcing of OSBL models is critically dependent on the horizontal resolution of the forcing, and somewhat on that horizontal resolution of the ocean model. As horizontal resolution is refined in an ocean model, sharp fronts can be resolved (Small et al., 2014) and thus the surface forcing will be highly variable in frontal regions. Therefore, it is expected that the behavior of OSBL models will vary based on resolution and model coupling frequency (e.g., Klingaman et al., 2011). In general, OSBL schemes aim to parameterize many important unresolved physical processes for use in ocean climate models. It is important to appreciate the limitations noted above concerning the space-time averaging scales. Providing a quantitative foundation for the space-time regime relevant for KPP or other boundary layer parameterizations is beyond our scope, but we provide examples of applicability for different vertical resolutions. Nonetheless, it forms an important problem in air-sea physics and numerical modeling that is becoming more timely as coupled climate and weather models evolve towards ever finer space-time resolutions. 3. Formulation of KPP for oceanic uses The implementation of KPP within CVMix closely follows that detailed in LMD94, including the modifications of Danabasoglu et al. (2006). We present here some salient points (a Table of important symbols and preferred units is given in Appendix A), aiming to elucidate important implementation considerations and to introduce some issues that arose during CVMix development. More details of potential problems are given in Section Elements of the KPP boundary layer scheme For any prognostic scalar or vector field component λ (e.g., velocity components, tracer concentrations), the KPP scheme parameterizes the turbulent 9

10 vertical flux within the surface boundary layer according to ( ) λ w λ = K λ z γ λ. (3) In this equation, the eddy diffusivity K λ is written as the product of three terms K λ = h w λ (σ) G (σ). (4) The boundary layer depth h > 0 scales the diffusivity, so that K λ is larger for deeper boundary layers. The non-dimensional shape-function, G(σ), is described in Section 3.2. The turbulent velocity scale, w λ 0, is computed according to ( ) κ u w λ =. (5) φ λ (σh/l) In this expression, κ = 0.4 is the von Kármán constant, u 0 is the friction velocity scale (determined by the square root of the surface stress magnitude), σ is the non-dimensional boundary layer coordinate (see equation (9) below), and L is the Obukhov length scale. The function φ λ 0 is a non-dimensional flux profile that is smaller for negative buoyancy forcing and goes to unity in the absence of buoyancy forcing. Given this form, the velocity scale w λ is larger for unstable surface boundary forcing (i.e., negative buoyancy forcing such as when removing heat or adding salt), as well as for stronger mechanical forcing (i.e., larger friction velocity scale as under strong wind forcing). The KPP vertical viscosity (used for frictional transfer of momentum in the ocean interior) is specified via a separate dimensionless flux profile through the Prandtl number ( ) ( ) Kv φλ P r = =. (6) K λ See Appendix B of LMD94 as well as Griffies et al. (2015) for full details of the non-dimensional flux profile functions φ λ and φ m, as well as the Obukhov length scale L. φ m 3.2. The non-dimensional shape function The vertical position in the ocean, z, ranges from H(x, y) z η(x, y, t) (7) 10

11 where z = H is the position of the static ocean bottom, z = η is the position for the dynamic ocean free surface, z = 0 is the resting ocean surface (reference geopotential), and depth = z + η 0 (8) is the positive distance from the ocean surface to a point in the ocean. When formulating equations within the boundary layer, it is convenient to use a non-dimensional boundary layer coordinate σ z + η h 0 σ 1, (9) where σ = 0 at the ocean free surface and σ = 1 at the boundary layer base (where z + η = h). In the KPP diffusivity expression (equation (4)), the non-dimensional shape function, G(σ), is assumed to take a polynomial form proposed by O Brien (1970) G(σ) = c 1 + c 2 σ + c 3 σ 2 + c 4 σ 3, (10) where c 1, c 2, c 3, c 4 are constants to be specified by the following considerations. First, since the diffusivity, K λ, is assumed to go to zero at the ocean surface, c 1 = 0. (11) Within the surface layer (0 σ ɛ) (see Figure 1), we can eliminate the gradient of λ in equation (3) using Monin-Obukhov similarity theory in the form ( ) λ z = (w λ ) sfc φ λ, (12) κ z u where (w λ ) sfc is the turbulent boundary flux crossing the ocean surface (e.g., turbulent latent and sensible heat; turbulent tracer flux; turbulent momentum flux). If we combine equations (3) and (4) and assume positive surface buoyancy forcing so that the non-local term vanishes (γ λ = 0) we have ( ) λ w λ = h G w λ, (13) z which returns us to the KPP closure form assumed in equation (3). Now insert equation (12) into equation (13), and assume G(σ) σ(c 2 + c 3 σ) 11

12 (valid in the surface layer where σ ɛ 1), to yield ( ) (w λ w λ = ) sfc φ λ h σ (c 2 + σc 3 ) w λ. (14) κ z u Using equations (5) and (9) brings equation (14) to the form ( w λ (w λ ) sfc ) = c 2 + σc 3. (15) Now we assume a linear decrease of the turbulent flux within the surface layer (i.e., (w λ )(ɛ) = β (w λ ) sfc where β is a constant), so that the surface flux at a position σ within the surface layer is given by ( ) w λ = 1 + σ (w λ ) sfc ɛ (β 1) = c 2 + σ c 3. (16) To be valid at σ = 0 requires c 2 = 1. (17) To determine the final two shape function coefficients, we require matching across the base of the boundary layer, at σ = 1. Use of the cubic equation (10) and its derivative at the boundary layer base leads to the following expressions ( ) G c 3 = G(1) (18a) c 4 = 1 2 G(1) + ( G σ σ ) σ=1. (18b) σ=1 Thus, the shape function is dependent on the chosen boundary conditions at the base of the OSBL. We next consider these boundary conditions Diffusivity matching for the shape function at the OSBL base LMD94 suggest that the diffusivity and viscosity predicted by KPP (equation (4)), as well as its vertical derivative, should match that predicted by the sum of all mixing parameterizations in the region below the boundary 12

13 layer (the ocean interior). To ensure appropriate matching, the necessary inputs to equations (18a) and (18b) are given by G(σ) = K λ INT (h) h w λ (σ) G σ = (19a) (h) + K ) λ INT (h) σ w λ (σ) w λ (σ) h wλ 2(σ), (19b) ( z K INT λ 277 where we evaluate terms on the right hand side at the boundary layer base, 278 σ = 1. This assumed matching means that parameterized turbulence gener- 279 ated in the ocean interior (e.g., internal waves, shear instabilities) indirectly 280 influences that in the surface boundary layer. Note that given that the inte- 281 rior diffusivity, Kλ INT, is arbitrary, equation (19b) constrains the magnitude 282 of the diffusivity in the OSBL, but not the sign. Yet, decaying turbulence 283 in the interior is not expected to influence mixing in the OSBL, thus we can 284 physically argue that equation (19b) should be constrained positive. These 285 points are relevant for numerical implementations, as now discussed. 286 In practice, the matching of diffusivities and their derivatives at the OSBL 287 base can lead to numerical problems. For example, diffusivity derivatives can 288 be sharp (i.e., discontinuous) near the OSBL base and can thus be poorly 289 represented on a discrete vertical grid. Furthermore, derivatives that are 290 reasonably well resolved can in fact cause negative diffusivities when higher 291 order interpolation is used (see Section 6.2). So in this case, poor resolu- 292 tion and/or linear interpolation artificially smooths the diffusivity profile, 293 mitigating the appearance of negative diffusivities. 294 Another option is for internal mixing schemes in the OSBL to be treated 295 additively with KPP, by assuming that both parameterize processes on dif- 296 ferent spatial and temporal scales, which disregards the view of Wyngaard 297 (1983) that boundary layers have distinct physics. With this assumption, 298 the boundary layer and interior parameterizations act in tandem within the 299 surface boundary layer; i.e., the sum of the KPP and interior diffusivities is 300 used within the surface boundary layer. Correspondingly, the KPP diffusiv- 301 ities smoothly go to zero at the OSBL base, so that we remove diffusivity 302 matching altogether. Without diffusivity matching, the shape function takes 303 the relatively simple cubic form used by Troen and Mahrt (1986) G (σ) = σ (1 σ) 2. (20) 304 In Section 6, we compare results using the original matching assumptions 13

14 (leading to equations (19a) and (19b)) to the simpler no match assumptions leading to equation (20). We return to diffusivity matching in Section Bulk Richardson number and the boundary layer depth In KPP, the bulk Richardson number is computed by Ri b = [b sl b(z)] ( z + η) u sl u(z) 2 + V 2 t (z). (21) In this expression, b is the buoyancy (dimensions of length per squared time), with b sl its surface layer average (averaged over the depth ɛ σ 0 shown in Figure 1). Small differences of b sl b(z) signal weak vertical stratification, characteristic of a region within the surface boundary layer. In contrast, large differences arise when z reaches into the more stratified region beneath the boundary layer. The denominator in Ri b consists of the squared vertical shear resolved by the model s prognostic velocity field: u sl u(z) 2, where u sl is the surface layer averaged horizontal velocity. In addition, the term Vt 2 aims to parameterize unresolved vertical shears near the OSBL base (see Section 4.2). When either the resolved or parameterized shear is large, the bulk Richardson number is small and the OSBL deepens. Finally, note that even if the buoyancy difference and vertical shear are vertically constant, the bulk Richardson number increases linearly with depth, d = z + η, given the presence of depth in the numerator of equation (21). The depth at the base of the ocean surface boundary layer, z + η = h, is computed as the depth where the bulk Richardson number equals to a critical value [b sl b(h)] h Ri b = Ri crit = u sl u(h) 2 + Vt 2 (z). (22) In LMD94, the critical bulk Richardson number was set to Ri crit = 0.3. However, values between 0.25 and 1.0 have been used in similar formulations (e.g., Troen and Mahrt, 1986; Vogelzang and Holtslag, 1996; McGrath-Spangler et al., 2015). In general, the correct diagnosis of the boundary layer depth is a key part of the KPP scheme, as this depth controls the upper ocean turbulent layer and the strength of mixing within that layer. The diagnosed boundary layer depth also controls the strength of entrainment into the OSBL. We thus expect the chosen definitions of surface layer fraction (ɛ; Figure 1), parameterized vertical shear, Vt 2, and the critical bulk Richardson number, Ri crit, to strongly influence KPP results and their comparison to LES. 14

15 Non-local tracer transport The non-local flux in KPP is the product of the diffusivity (equation (4)) and a parameterized non-local term. The non-local term is non-zero only for tracers in unstable forcing (though see Smyth et al. (2002) who suggest a form for momentum), and it takes the form (e.g., Mailhot and Benoit 1982, LMD94) ( ) γ λ = C κ (c s κ ɛ) 1/3 (w λ ) sfc, (23) w λ h where κ is the von Karman constant and c s and C are constants (see LMD94 or Griffies et al. (2015) for details of these constants). The turbulent flux (w λ ) sfc arises from surface tracer transport due to air-sea or ice-sea interactions. Thus the non-local potential temperature flux (λ = θ), which is the product of the diffusivity and the non-local term, is given by K θ γ θ ( w θ ) non-local = C κ (c s κ ɛ) 1/3 G(σ) (w θ ) sfc. (24) In this form, we see that the KPP non-local potential temperature flux is a vertical redistribution of the surface boundary flux (w θ ) sfc throughout the boundary layer. The surface potential temperature flux includes penetrating shortwave radiation. However, it is unclear how much of the shortwave absorbed in the boundary layer to include in (w θ ) sfc. The choice is important as it impacts on the strength of the non-local term. For example, use of all of the incident shortwave in (w θ ) sfc results in smaller magnitude for the destabilizing buoyancy forcing and thus a smaller non-local term. We have more to say regarding the non-local tracer flux formulation in Section KPP implementation considerations The LES simulations defined in Tables 1-2 have exposed a number of key considerations in the formulation of KPP as implemented by CVMix, some of which have been noted in part by LMD94 (their Appendix D) and Large and Gent (1999). We summarize these considerations in this section Determining the OSBL depth Determination of the ocean boundary layer depth in KPP according to equation (22) is dependent on three key parameters: the critical Richardson number, Ri crit, the parameterized shear, Vt 2, and the fraction of the boundary layer that is within the surface layer, ɛ (see Figure 1). Values of Ri crit vary 15

16 from Ri crit = 0.25 for shear instability when using a linear stability analysis (Miles, 1961) 3 to O(1) for a non-linear stability analysis (Abarbanel et al., 1984). However, Troen and Mahrt (1986) argue that shear may not be adequately resolved in a model simulation, prompting use of a larger value of Ri crit that is a function of vertical grid spacing. Numerous OSBL depth formulae have been tested (Vogelzang and Holtslag, 1996; Seibert, 2000; Seidel et al., 2010, 2012; McGrath-Spangler and Molod, 2014; McGrath-Spangler et al., 2015; McGrath-Spangler, 2016). Vogelzang and Holtslag (1996) found that a boundary layer depth formula similar to that in KPP accurately predicted observed depths (based on a TKE threshold criterion) in stable and near-neutral conditions. Yet McGrath- Spangler and Molod (2014) note that an OSBL depth formula depending on diffusivity results in slightly deeper depths relative to bulk Richardson methods (as in KPP). McGrath-Spangler et al. (2015), McGrath-Spangler (2016), and Seidel et al. (2010) found differences between OSBL depth formulations on the order of 20%. These studies were unable to conclusively determine an ideal OSBL depth diagnostic due to insufficient observational data. In the bulk Richardson number (equation (21)), the shear and buoyancy differences are relative to the buoyancy and momentum carried by the parameterized boundary layer filling eddies. Hence, the chosen surface layer extent, ɛ, will influence the OSBL depth (this point was also raised by Seidel et al., 2010). Further, it is unlikely that the bulk Richardson number computed at a model interface is exactly equal to the chosen critical threshold. Consequently, the KPP boundary layer depth, h, will also be sensitive to the interpolation method used to determine where Ri b = Ri crit according to equation (22) (Danabasoglu et al., 2006; Seidel et al., 2010). Notably, in our comparisons between KPP and LES, we use the KPP bulk Richardson number method based on equation (22) to determine the OSBL depth with LES data The parameterization of Vt 2 in Ri b The KPP scheme includes a term related to an energy associated with unresolved turbulence (Vt 2 ) in the bulk Richardson number denominator in equation (21). The purpose of this term is to sufficiently deepen the OSBL to 3 This definition may not be appropriate for weak mean shear and breaking internal waves (Troy and Koseff, 2005; Barad and Fringer, 2010). 16

17 ensure that the empirical rule of convection is satisfied. This rule says that the minimum turbulent buoyancy flux within the boundary layer is given by (w b ) he 0.2 (w b ) sfc, (25) where h e is the depth of the minimum buoyancy flux (the entrainemnt depth, see Figure 2). The parameterization of Vt 2 is derived by considering a buoyancy profile that is well-mixed to a given depth (h m ) with linear stratification below (Figure 2). The buoyancy flux at h e is written (using equations (3) and (4)) as ( ) ( w b ) b h e = h w s (σ) G(σ) z γ b with σ = h e h. (26) Near the boundary layer depth, h, the non-local term γ b (equation (23)) is small and can be ignored. Furthermore, in convective conditions, w s (h e /h) κ (c s κ ɛ) 1/3 w, (27) 409 where w is the convective turbulent velocity scale, which is defined as w [ h (w b ) sfc ] 1/3. (28) 410 Also, noting that N 2 = b/ z and h e /h 1, equation (26) becomes ( w b ) h e = κ (c s κɛ) 1/3 w (1 h ) 2 e h N 2 h = κ (c s κɛ) 1/3 w (h h e ) 2 N 2 /h. (29) At the OSBL base, Ri b = Ri crit (equation (22)). We can use this result to write an equation for Vt 2, and by assuming the resolved vertical shear is zero Ri crit = h [b sl b(h)]. (30) Vt 2 Given the linear buoyancy profile, the numerator of equation (30) is h N 2 (h h e ), which implies ( ) V 2 h h e = t Ri crit. (31) h N 2 17

18 Figure 2: Schematic illustrating the various depths arising in the boundary layer as forced by a destabilizing surface buoyancy flux (w b ) sfc < 0 (defined in equation (50)). This forcing supports active non-local convective boundary layer turbulence. Data for this figure is taken from the LES results of the Convect I experiment in Section 6.1. The vertical axis is the non-dimensional boundary layer depth, σ = ( z + η)/h (equation (9)), extending from beneath the boundary layer (σ > 1) to the base of the surface layer (σ = ɛ h) (see Figure 1). The dashed line is the local turbulent buoyancy flux, (w b ), normalized by the surface buoyancy flux (w b ) sfc. The scale for this ratio is along the top axis. Depths where (w b )/(w b ) sfc < 0 are where the local turbulence stabilizes the boundary layer, which occurs near the boundary layer base. The solid line is the difference in local buoyancy and the surface layer buoyancy, b b sfc, with corresponding scale along the bottom axis (in units of 10 4 m s 2 ). The mixed layer is where buoyancy is weakly stratified, b/ z 0. The mixed layer base is determined subjectively by b/ z > ( b/ z) min > 0, with ( b/ z) min a chosen minimum stratification criteria. For much of the boundary layer, b b sfc > 0, since the surface layer buoyancy is driven low by the destabilizing surface flux. The entrainment depth, h e, is where (w b )/(w b ) sfc < 0 reaches a minimum, with h e straightforward to diagnose in a LES. (See Appendix D for an idealized buoyancy profile similar to that shown here allowing for an analytic expression for h). In the entrainment layer (where water from below the boundary layer base is exchanged with boundary layer water), buoyancy changes 18 rapidly ( b), reflecting enhanced vertical stratification below the mixed layer. Below the entrainment layer, the buoyancy profile is roughly constant (b N 2 z with N 2 > 0 constant and z < 0). The KPP boundary layer depth, h, is determined by the bulk Richardson number criteria in equation (22).

19 415 Using equations (30) and (31) brings equation (29) into the form ( w b ) h e = κ (c s κɛ) 1/3 w V 4 t (Ri crit ) 2 N 2 /(N 4 h 3 ). (32) If the empirical rule of convection ( ) (w b ) he 0.2 (w b 416 ) sfc is enforced in equation (32), the result can be solved for V so that t V 2 t = 0.2 Cv Ri crit κ 2/3 (c s ɛ) 1/6 h N w (33) where LMD94 included the constant C v to account for smoothing of the buoyancy profile near h e due to mixing. Note here that N is the stratification near h e < h, which is counter to using stratification below the OSBL suggested in Danabasoglu et al. (2006) (their Appendix A). Both options are tested relative to LES (section 6.1.2) Shape function and diffusivity matching As discussed in Section 3.3, matching diffusivities across the OSBL base can result in significant numerical error, particularly when diffusivities have rapidly changing vertical derivatives. This problem can be addressed in two ways. First, the sign of the interpolated diffusivity and corresponding vertical gradient at the OSBL base could be constrained to be positive. Alternatively, diffusivity matching could be neglected, in which case the shape function takes the form given by equation (20). As a physical justification for dropping the diffusivity matching, one may assume that KPP is only parameterizing effects from the larger and stronger turbulent boundary layer eddies. Other parameterizations (e.g, the LMD94 shear instability mixing scheme) are assumed to be scale separated from KPP as KPP seeks to parameterize the penetration of the most unstable eddies. Under this assumption, the net diffusivity in the OSBL is computed as the sum of that predicted by KPP plus other parameterizations. The chosen shape function also exerts a strong influence on the upper ocean heat flux. If the KPP computed boundary layer diffusivities (and the derivatives) are forced to match the corresponding interior values, G(σ) will not be zero at the base of the boundary layer. A non-zero shape function at the OSBL base is not desirable, as the non-local flux in equation (24) parameterizes boundary layer filling eddies and the flux should become zero at the boundary layer base. There are two possible resolutions to this inconsistency: (A) have different shape functions for the down-gradient and the 19

20 non-local components of the parameterized tracer flux, or (B) neglect diffusivity matching and thus utilize the shape function defined in equation (20). There are reasons to consider a combination of options (A) and (B), due to the non-local tracer transport, which are discussed in Section Non-Local Transport We introduced the KPP parameterization of non-local tracer flux in Section 3.1. We summarize here some specific issues related to its formulation and calculation Salt flux LMD94 formulated KPP for models with a virtual salt flux. In this case, the non-local salt fluxes are directly implemented just as the non-local heat fluxes given by equation (24). In models making use of a freshwater surface boundary condition rather than a virtual salt flux 4, salinity in the top layer is changed via surface mass fluxes (Q m arising from precipitation, evaporation, runoff, ice melt/formation). Implementation of the non-local transport in this framework requires a separately defined salt flux given by Q s = Q m S sfc, (34) where S sfc is the surface salinity. Equation (34) also applies to all tracers with high solubility that do not leave the ocean with evaporating water Shortwave radiation The non-local heat flux as defined by LMD94 (equation (24)) is non-zero only for negative surface buoyancy forcing. There are cases where negative surface buoyancy forcing can still contain a component from shortwave radiation (unless under sea ice and/or in the polar winters). The penetration of shortwave radiation into the ocean is distinct from other heat fluxes. It therefore requires separate considerations for its treatment in the non-local heat flux. The degree to which shortwave radiation penetrates into the ocean depends on optical properties of seawater (e.g., Paulson and Simpson, 1977; Manizza et al., 2005). At the depth of absorption, incident photons add heat 4 In this paper, MPAS-O and MOM use natural boundary conditions where fresh water is exchanged at the ocean-atmosphere interface, whereas POP uses a virtual salt flux. 20

21 to the ocean. In the presence of negative surface buoyancy forcing, a portion of the absorbed heat is carried deeper into the boundary layer through both local and non-local boundary layer turbulence. Thus the shortwave radiation contribution to the surface buoyancy flux should be carried in the parameterized non-local heat flux. However, it is not clear just what portion of total shortwave radiation entering the ocean surface is to be carried as part of the non-local term. When the salinity exceeds 25 ppt, the lowest temperature is the temperature of maximum density (freezing point), so that shortwave heating stabilizes the water column. In this case, the impacts of adding shortwave radiation to the non-local flux have been found to be minimal (see Figure C4 of LMD94). The reason is that where S > 25 ppt, shortwave heating does not often coincide with destabilizing buoyancy fluxes and the non-local parameterization will not be active. In contrast, in many coastal ocean regions with fresh water, the lowest temperature does not correspond to the maximum density. Thus for brackish waters, for example near melting sea ice, shortwave radiation can be destabilizing since it increases ocean density. The treatment of water column destabilization by a large portion of short wave radiation is discussed below in Section Derivation of the non-local transport parameterization The non-local flux is commonly considered a correction to local downgradient diffusion, where the tracer gradient vanishes (e.g., Deardorff, 1966; Troen and Mahrt, 1986). With this interpretation, one may be motivated to set the shape function for the parameterized non-local flux the same as that for the parameterized downgradient flux. However, this approach is subject to caveats that break the connection between the local and non-local shape functions. To expose the caveats, we revisit the derivation of the non-local transport. To derive the non-local parameterization, consider the potential temperature flux equation after assuming steady flux, ignoring horizontal processes, and small viscosity effects (e.g., Donaldson, 1972), (w w θ ) z = θ ρ o p z w 2 ( ) θ ) + g (α θ θ z 2 β S θ S. (35) The third-order term on the left hand side is the vertical turbulent transport of the turbulent potential temperature flux. The first term on the right 21

22 side is the temperature pressure covariance. The second term is the local generation of the temperature flux, and the final two terms arise from buoyant production. In many cases the pressure temperature covariance (first term on right hand side of equation (35)) can be approximated by a return to isotropy form (Rotta, 1951). The return to isotropy assumption represents an exchange among directional components of the heat flux that affect a return to an isotropic flow state (i.e., directional invariance). A number of studies (e.g., Moeng and Wyngaard, 1986; Canuto et al., 2001) find that buoyancy and shear contributions are needed in the parameterization of the pressure scalar covariance. Yet in highly convective conditions the shear contribution is minimal and the pressure temperature covariance is approximated by θ p ρ o z θ ( ) w g d 1 α θ θ τ 2 β S θ S, (36) where τ is the return to isotropy time scale and d 1 is a constant. With this assumed parameterization, equation (35) becomes ( ) w θ θ = w τ 2 (w w θ ) ) + g(1 2d 1 ) (α θ θ z z 2 β T θ S. (37) In equation (37), the first term on the right is the local contribution and the remaining terms are the non-local terms. Holtslag and Moeng (1991) (hereafter HM91), using atmospheric LES results suggest that d 1 0.5, which eliminates the buoyant production term. Yet, Canuto et al. (2007) suggest a different value of d 1 and the buoyant production terms remain. KPP, following Deardorff (1972), assumes that the vertical turbulent transport of the heat flux, (w w θ )/ z, is small relative to the buoyant production terms (e.g., see Figure 10). The buoyant production term (assuming θ S is small) is parameterized via Monin-Obukhov similarity theory (e.g., Mailhot and Benoit, 1982; HM91). With these assumptions, the vertical turbulent temperature flux in equation (37) becomes ( θ w θ τ w 2 z a ) 1 w (w θ ) sfc (38) h w 2 where a 1 is a constant, and w is the convective velocity scale, defined in equation (28) above, and h is the boundary layer depth. The assumed scalings for w 2 and θ 2 in equation (38) are only valid for strongly surfaced forced 22

23 cases (HM91). When surface forcing of a given tracer (e.g., salt or a nutrient) is small and yet the temperature flux is destabilizing, assuming that the nutrient variance scales with the surface flux is not appropriate (Section 6.2; see also Moeng and Wyngaard, 1984; Noh et al., 2003). If we assume that the vertical velocity variance scales with the convective velocity scale, then the non-local term is written as γ θ = a 1 a 2 (w θ ) sfc h w. (39) The term a 1 /a 2 assumes the role of C in KPP and similar models (compare to equation (23) where equation (27) is used to convert w to w s ). LMD94 assumed that C 10 in highly convective conditions. However, results from HM91 and our simulations (not shown), suggest a value between 2.5 and 5 (see HM91, their Figure 2) Concerning the choice of a shape function If equation (38) is utilized to parameterize non-local transport, the same shape function is used for downgradient and non-local portions of the parameterized temperature flux. However, multiplying τ w 2 through the parenthesis in equation (38) reveals that the parameterized non-local temperature flux is written as (w θ ) non-local = a 1 τ w (w θ ) sfc. (40) h With equation (40) to parameterize the non-local contribution to the heat flux, the shape function would be contained in the return to isotropy time scale (τ) for the non-local transport term. Given that τ w 2 assumes the role of h w s (σ) G(σ) in KPP, we might expect that the shape function chosen for the separated non-local term should be related to that chosen for the downgradient diffusivity, due to τ multiplying both local and non-local terms. However, there is no fundamental requirement that the shape functions be identical, since the local and the non-local terms describe processes of different physical nature. The cubic shape function chosen in KPP, and similar models (e.g., Troen and Mahrt, 1986; Holtslag and Boville, 1993) follows from O Brien (1970). O Brien (1970) chose a cubic shape function for eddy diffusivity and imposed a few physical requirements. The shape function and its derivative must be continuous. 23

24 The shape function and its derivative must match similarity theory at the bottom of the surface layer. The diffusivity is small at the boundary layer base. These conditions also imply that the shape of diffusivity is set by properties of the surface layer alone despite an expectation that boundary layer quantities (e.g., stability, current shears) would also influence diffusivities. There is also no physically motivated argument for a universal cubic shape function. Indeed, Lettau and Dabberdt (1970) argue from theoretical and observational constraints that a parabolic shape function for diffusivity is required when the angle formed by the shear stress and mean wind departure from geostrophy is invariant with height (e.g., Ekman spiral). When this condition does not hold, Lettau and Dabberdt (1970) suggest that a cubic shape function matches their limited data Considerations regarding the non-local heat flux parameterization The parameterized non-local tracer transport is one of the defining elements of the KPP scheme. However, its use comes with some caveats regarding the potential for unphysical behavior. We articulate some of the problems here. There are some issues that emerge when choosing the cubic shape function G(σ) (equation (10)) in the parameterized non-local heat transport. First, when assuming a cubic shape function in the non-local heat flux parameterization, the upper 1/3 of the OSBL warms and the lower 2/3 cools due to the sign change in G(σ)/ σ. This character of the heat flux can lead to unphysical behavior. For example, consider constant surface cooling from a sensible heat flux, and assume the column is stratified in salinity but vertically constant in temperature. Assume no penetrative shortwave radiation and no surface water fluxes. In this case there is no baroclinicity and the surface forcing creates no resolved vertical shear. The surface cooling thus imparts a negative (destabilizing) buoyancy forcing, which means the KPP non-local heat transport parameterization is active. Initially, OSBL mixing penetrates to a depth determined by the salinity stratification. At these early times, local diffusion of temperature is minimal since θ/ z 0. Despite diffusion having not yet reached this depth, the instantaneous effects of the non-local parameterization warm depths in the upper 1/3 of the OSBL. The result is upper boundary layer warming in the presence of surface cooling. Eventually, the 24

25 local diffusive parameterization spreads the cooling throughout the OSBL to thus remove the unphysical warming present early in a transient response to surface cooling. In a related example, consider low salinity waters (S < 25 ppt) in the presence of shortwave radiation. In such brackish waters, shortwave heating can be destabilizing (i.e., heating reduces buoyancy), if the SST is below the temperature of maximum density. If all of the incident radiation is transported by the parameterized non-local heat flux, sea-ice could be formed in the presence of surface heating. This response is unphysical. Given the uncertainty in surface flux formulations (see Section 2.2) it could be argued that these pathological examples are the result of considering the short-time transient phase of a simulation, whereas the parameterization is only expected to perform physically when averaged over a longer time period. However, the pathology can arise in realistic simulations and it can lead to non-linear responses (e.g., more sea ice growth due to shortwave heating). We see examples of such pathologies in Convect I and Diurnal I (Sections 6.1 and 6.4). The problems therefore must be addressed for general purpose use of KPP Issues regarding the non-local tracer flux parameterization Issues associated with the KPP parameterized non-local flux also arise when considering a case with no passive tracer in the OSBL and a nonzero surface flux that adds tracer to the ocean. The cubic shape function will create a negative concentration of this tracer in the upper 1/3 of the OSBL. In realistic model configurations, the time evolution of scalar fields may be dominated by other processes (e.g., advection and horizontal mixing) that may mask these unphysical effects. Nonetheless, negative concentrations for material tracers, particularly biological tracers, can have severe adverse effects on a simulation Possible remedies for the non-local tracer flux parameterization issues The above problems with the non-local parameterization can be mitigated numerically or physically. For example, LMD94 utilized a semi-implicit time-stepping scheme, requiring multiple iterations per time step (see their Appendix D). The LMD94 scheme may not develop artificial stratification, as it is could be mixed out during the iterative time-stepping procedure. Alternatively, the physical assumption within KPP of a cubic shape function could be relaxed for the non-local redistribution of scalar fluxes, while 25

26 maintaining the original cubic shape function for local diffusivity. The new non-local shape function must meet two requirements: Monotonic, which eliminates the spurious warming in the presence of surface cooling and the negative tracer concentrations for initially uniform concentration profiles, and Vanish at the OSBL base (σ = 1). To ensure monotonicity of the non-local shape function, we can no longer require that it vanishes at the ocean surface. However, a vanishing shape function is required for the downgradient portion of the KPP flux parameterization (equation (3)). Hence, we are led to separate shape functions for the two portions of the KPP flux: G(σ) for the downgradient portion, and G(σ) for the non-local portion. We here discuss constraints for the non-local shape function. If we consider the non-local parameterization to be a redistribution of the surface tracer flux (see equation (24)), we may assume that the parameterized non-local flux at the ocean surface, σ = 0, equals to the turbulent surface flux. Making use of this assumption in equation (24) leads to w θ non-local(σ = 0) = (w θ ) sfc = C s G(0) (w θ ) sfc G(0) = Cs 1, (41) 656 where we introduced the shorthand C s = C κ (c s κ ɛ) 1/3. (42) 657 With the above requirements, CVMix has implemented the following G(σ) = C 1 s (1 σ) 2 (43) as an option within the non-local tracer flux parameterization. This shape function is monotonic within the OSBL. Hence, it does not exhibit spurious heating in the presence of cooling discussed above, and it eliminates the potential for negative passive tracer concentrations. Note that this option is just one of many possible functions that are monotonic in the OSBL and vanish at the OSBL base. When equation (43) is utilized, the surface flux cannot be applied to both the local and non-local portions of the tracer flux. If the surface flux is applied 26

27 to tracers and then KPP is invoked with equation (43), then the flux would be applied twice. Thus when equation (43) is utilized in model simulations, we set G(0) = 0 so the calling model can easily utilize equations 43. The implementation remains monotonic as the application of the surface flux by the calling model will apply equation (41). Equation (43) has been implemented is tested against LES in a number of our test cases. Note, that the implementation of equation (43) in CVMix/KPP assumes the no-match configuration (Section 3.3) Grid resolution and mixing near the boundary layer base Mixing at the boundary layer base results in the transport of properties across the boundary layer base. Therefore, it is expected that mixing and vertical grid resolution near the OSBL base will greatly influence the boundary layer properties. In their Appendix D, LMD94 note that as vertical resolution coarsens near the OSBL base, stair case structures can emerge in time series of the boundary layer thickness and its properties. While use of quadratic interpolation to determine the OSBL depth has reduced these stair case structures (Danabasoglu et al., 2006), we still see such structures in our simulations (e.g., Figure 4a). We thus summarize here the reasons for this behavior and offer remedies based on Appendix D from LMD94. Start by assuming the OSBL base is aligned with a grid cell base. Furthermore, as in the case discussed in LMD94 (and test case Convect I described in Section 5.2), assume zero vertical property gradients within the boundary layer. Now allow the OSBL to deepen further into a new grid cell. As it deepens, boundary layer induced mixing within the new grid cell is not possible until the boundary layer reaches the bottom of this new cell. Once the OSBL depth reaches down to the next model interface, diffusivities become large, in which case properties mix quickly. The resulting time series of boundary layer properties thus exhibits a stair-step structure. As a remedy to the stair-step structure, LMD94 propose the addition of an enhanced diffusivity at the OSBL base. This diffusivity allows for property exchange across the boundary layer base, thus acting to smooth the boundary layer s vertical movement between grid cells. The enhanced boundary layer diffusivity is computed as K x = (1 δ) 2 K(k osbl 1/2 ) + δ 2 K(k osbl ). (44) In this equation, K(k osbl 1/2 ) > 0 is the KPP boundary layer diffusivity interpolated to the center of the grid cell containing the boundary layer 27

28 base, with k osbl the discrete index for this grid cell base and k osbl 1/2 the corresponding index for the cell center (k increases downward; see Figure 3). The diffusivity K(k osbl ) is at the base of this grid cell. K(k osbl ) is zero when using the no-matching conditions discussed in Section 3.3, or non-zero when matching diffusivities across the boundary layer base to the non-zero values beneath the boundary layer. The fractional distance from the last grid center in the OSBL, δ, is given by δ = h z(k osbl 1/2), (45) z where z z(kosbl+1/2 ) z(k osbl 1/2 ) 708. Equations (44)-(45) were determined empirically. The variables in equation (44) along with δ are shown in Figure The enhanced KPP diffusivity at the OSBL base is then given by Λ x = δk x. (46) The resulting enhanced diffusivity for the case in Figure 3 is given by the black diamond. The increased property exchange across the boundary layer base acts to reduce the vertical grid resolution dependence of KPP, with examples shown in Figure 5 of this paper and Appendix C of LMD94. As the grid spacing near the OSBL is refined, K(k osbl 1/2 ) 0, assuming zero background diffusivity and/or zero diffusivity matching across the boundary layer base, the influence of enhanced diffusivity decreases. This behavior suggests that the need for enhanced diffusivity could be alleviated when using a hybrid vertical coordinate that chooses a fine vertical resolution near the OSBL base. 5. The LES model and the one-dimensional test cases In this section, we present the LES model used for establishing a baseline behavior to be compared with KPP. We then summarize the one-dimensional oceanic test cases used to benchmark KPP. A description of the LES tests is summarized in Table 1-2. The LES tests cover a range of parameters aiming to examine the potential issues described in Section Large Eddy Simulation (LES) model We compare one-dimensional column tests of KPP to a mature and commonly utilized LES model (Moeng and Wyngaard, 1984; McWilliams et al., 28

29 Figure 3: Schematic illustrating the enhanced diffusivity parameterization from LMD94 used at the base of the boundary layer. The solid circles are diffusivities computed by KPP at model grid interfaces. We assume here that K(k osbl ) is zero, as occurs for the no-matching conditions discussed in Section 3.3, or if matching to zero diffusivity beneath the boundary layer base. The solid black line is an assumed analytic representation of the diffusivity. The fractional distance of the OSBL depth (h) from the last model center in the OSBL is δ (equation (45)), whose vertical extent is given as δ z. The solid diamond at 100 m is the resulting enhanced diffusivity computed from equations (44) and (46). 29

30 1997; Sullivan et al., 2007). The three-dimensional, non-hydrostatic Boussinesq LES model equations are u t + (ω + f) u L = (p + 12 ) u L 2 + b ẑ, (47a) b t + u L b = 0 u = 0, (47b) (47c) where u is the Eulerian velocity averaged over surface gravity waves, ω u is the Eulerian vorticity, f is the Coriolis parameter, p is the pressure, b is the buoyancy, u L u + u s is the Lagrangian velocity, and u s is the Stokes drift velocity. We do not consider surface waves here, so that u s = 0 in all our tests. We have made two key modifications to the LES model for use in our tests. First, we include salinity by setting the buoyancy equal to b = g [ 1 α θ ( θ θ ) + βs ( S S )]. (48) In this equation, α θ is the thermal expansion coefficient, β S is the haline contraction coefficient, and the overline represents a horizontal average. For our tests, we choose the constant values α θ = K 1 β S = ppt 1. (49a) (49b) In addition, the buoyancy production terms in the sub-grid TKE scheme have been modified so that w b = g ( α θ w θ β S w S ). (50) Next, the stability factor in the length scale parameterization (Deardorff, 1980) has been modified, with evaluation of these changes presented in Appendix B. An additional modification involves the implementation of a diurnal cycle in the LES model along with shortwave radiation (similar to Wang et al., 1998). The vertical penetration of incoming solar radiation follows a twoband exponential formulation with constant extinction coefficients of a Jerlov type IB water mass (Paulson and Simpson, 1977). The function describing the time variation of surface shortwave radiation is described in Appendix C. 30

31 Description of the test cases Both transient and equilibrium test simulations are considered. The transient simulations are forced with a net heat loss, thus inducing convection and testing the non-local portion of the KPP closure. In all test cases, destabilizing fluxes (i.e., cooling, evaporation, and wind) are taken as constant in time. In the equilibrium experiments, the shortwave heat flux is constructed such that the daily integrated positive (stabilizing) buoyancy input is balanced by the daily integrated negative (destabilizing) flux. The derivation of equation (C.1) (see Appendix C) does not include the influence of wind and thus simulations such as Diurnal II are not in true equilibrium. For each diurnal run, the maximum shortwave radiation is calculated using equation (C.3) (see Appendix C) and is shown in Table Summary of the tests All tests are configured with both the one-dimensional KPP model and a horizontally homogeneous LES model. A summary of the test cases considered in our evaluation is as follows: parameter choices: The influence of parameter choices is made via variation of the critical Richardson number, Ri crit (see equation (22)), and the surface layer extent, ɛ (see Figure 1). Previously chosen values of Ri crit for KPP and similar schemes (e.g., Troen and Mahrt, 1986) have a wide range. Here ɛ is varied between 0.05 and 0.2, with no value specified from theory (the only specification is that ɛ 1). Notably, changing the value of ɛ impacts directly on the strength of the parameterized non-local flux (see equation (24)). Further, the chosen surface layer fraction (ɛ) influences the reference buoyancy in equation (22) and hence the OSBL depth. interpolation to determine depth of the boundary layer: Upon determining the grid cell Ri b > Ri crit, the boundary layer depth, h, is found by interpolating between model grid levels. Seidel et al. (2012) and Danabasoglu et al. (2006) have found that the diagnosed OSBL depths are very sensitive to the chosen interpolation for the surface layer averages and interpolation between layers. Interpolation techniques are primarily examined in the context of diffusivity matching (Section 3.3). 31

32 Simulation Qo(W m 2 ) Q max sw (W m 2 ) E (mm/day) τx(p a) T (z), S(z) Convect I A Convect II C Convect III A Convect Salt B Diurnal I A Diurnal II A Table 1: Summary of forcing scenarios considered in the test cases. Qo is the non-solar surface heat flux (positive is into the ocean), and Q max sw is the maximum of surface diurnal shortwave radiation. We provide details of the diurnal shortwave forcing in Appendix C. E is the surface evaporation rate, τx is the zonal wind stress, and τy = 0 in all simulations. T (z) and S(z) are the initial temperature and salinity profiles, which are given in Table 2. The Convect Salt test is used to verify the LES implementation of salinity Appendix B. 32

33 Profile T (z) S(z) A z 35 B z C 20 if z (z + 25) else 35 if z else Table 2: Initial temperature ( C) and salinity (ppt) profiles used in the experiments given in Table 1. Recall that the vertical position (geopotential coordinate z) is positive upwards, with z = 0 at the resting ocean surface and z < 0 in the ocean interior shortwave and the non-local transport: As discussed in Section 3.5, it is unclear how much shortwave radiation should be included in the KPP non-local heat transport. We examine this issue by considering both the shortwave absorbed in the top model layer and the full incident shortwave radiation absorbed in the OSBL Specifications for the tests A list of the various specifications for the test simulations is as follows: We use a linear equation of state (equation (48)). The values of α θ and β S used in equation (48) are identical to those used for the LES (equations (49a) and (49b)). Given a linear equation of state, the surface buoyancy flux is given by (w b ) sfc = g ( α θ (w θ ) sfc β S (w S ) sfc ). (51) A variety of vertical grid spacing is considered for the KPP tests: some with uniform vertical spacing (1 m to 10 m) and one with non-uniform grid spacing (1.5 m near surface). The LES utilizes a stretched grid over its 150 m vertical extent, with the first layer thickness of 0.1 m. The LES has a horizontal extent of 128 m and a uniform horizontal resolution of 0.5 m. All simulations in Table 1 are initialized with zero momentum. Temperature and salinity profiles for these simulations are shown in Table 2. The baseline KPP configuration for the experiments follows LMD94 and Danabasoglu et al. (2006), with a summary here given in Table 3. 33

34 run duration OSBL interpolation timestep interpolation order to determine κ int, ν int at σ = 1 shape function for κ and ν shape function for non-local internal matching convective diffusion default value 8 days quadratic 20 min linear LMD94 cubic LMD94 cubic LMD94 internal matching enabled Ri crit 0.25 ɛ 0.1 enhanced diffusivity at OSBL base shear instability mixing background diffusivity internal wave mixing double diffusion enabled zero zero zero zero Table 3: Summary of default or baseline KPP configuration. Note that in all runs, a large vertical diffusivity meant to parameterize convection is computed after KPP and only below the OSBL. 34

35 In the simulations Convect I, II and Diurnal I, all internal mixing schemes (mixing below the boundary layer) are disabled, so the diffusivity vanishes beneath the boundary layer. Hence, any diffusivity matching (Section 3.3) from LMD94 has no influence. In contrast, the experiments Convect III and Diurnal II enable interior shear instability mixing. We make use of the LMD94 shear mixing parameterization for these tests. The shape function is computed according to equations (18a)-(19b). The surface buoyancy flux, (w b ) sfc, in the KPP boundary layer algorithm only considers shortwave absorption in the boundary layer. The sensitivity to this choice is tested in Diurnal I and II. Across all configurations and parameter settings more than 100 tests were conducted. 6. Test case results and analysis In this section, we present results from the test cases outlined in Section 5.2, with further details also provided in the appendices. Each simulation is discussed with particular focus on physical and numerical issues introduced in Section 4. Note that some test cases expose more than one issue. Furthermore, numerous sensitivities are examined for each test. Table 4 summarizes the labels for each sensitivity test and changes made to KPP relative to the baseline configuration of Table Convect I In Convect I, we apply a destabilizing (cooling) surface boundary heat flux. There is no evaporative flux and salinity is vertically uniform (Table 2). This test illustrates some of the vertical grid resolution dependence of KPP (e.g., Figure 4a), along with further noisy behavior shown in Figure 6, discussed below. We offer recommendations to mitigate these issues in Section Baseline Results In Figure 4a, we show the OSBL depths from the baseline configuration (defined in Table 3) for KPP from MPAS-O, POP, and MOM. We also show results from the LES and the analytic solution derived in Appendix D (see 35

36 Test label Parameter(s) changed Base follows Table 3 NM PNL internal matching disabled G(σ) = C 1 s (1 σ) 2 as in equation (24) N2 N = max(n(k osbl 1 ), N(k osbl )) as in equation (55) Match Q MatchLimit Match Diff compute K int λ require K int λ (h) and zk int (h) with quadratic interpolation λ (h) 0 and zk int (h) 0 for internal matching λ assume z K int (h) = 0 in internal matching λ Table 4: Sensitivity tests conducted. Test labels correspond to figures in Section equation (D.10)). The LES and analytic solution show consistent behavior. Likewise, results from MPAS-O, POP, and MOM are consistent across the three models. Surprisingly, the coarser grid resolution computes a KPP boundary layer depth consistent with the LES, whereas the finer grid resolution simulations exhibit a persistent shallow boundary layer bias. This underestimation of mixing at the OSBL base is discussed in Section We also note, that near the beginning of the simulation high-frequency temporal noise in the simulated OSBL depths is evident in all three models (Figure 4a). This noise is likely due to the artificial stratification caused by the cubic shape function in the non-local flux parameterization (Section 4.4.5). The high-frequency noise is further discussed in Sections and 6.4. Figure 4b shows the time averaged, normalized buoyancy flux profiles. The fluxes are very sensitive to the averaging interval. This is not surprising given the stair case structures evident in Figure 4a. When the OSBL deepens rapidly, the buoyancy flux minimum is larger and decreases as the rate of OSBL deepening weakens. As resolution increases, the variability in the buoyancy flux near the OSBL base weakens. To account for the buoyancy flux variability at coarse resolution we have averaged over the rapid deepening near the beginning of day 6 in Figure 4b. Consistent with the OSBL depths, in the high resolution KPP simulations the depth of the buoyancy flux minimum (h e from Figure 2) is shallower than LES and coarse resolution results. Additionally, the magnitude of the 36

37 entrainment flux at high resolution is too small, which is consistent with the shallow OSBL bias relative to LES. Time averaged buoyancy profiles (Figure 4c) show biases consistent with Figure 4a-b. The high resolution profiles are under-mixed relative to coarse resolution and LES. The entrainment layer depth ( 60m depth, region of large stratification) is also too shallow at high resolution High Resolution Bias To understand the unexpected bias in the fine resolution KPP simulations, we reexamine how the boundary layer depth is predicted in KPP. For Convect I, equation (22) simplifies to Ri crit = h [b sl b(h)] C h N osbl w. (52) To reach this expression, we made use of a simplified definition of Vt 2 in which all the constants in equation (33) have been subsumed into C, and we assume u, v = 0. In Convect I, the only variable in equation (52) that can potentially increase OSBL depths at fine resolution is stratification near the OSBL (N osbl ). Recall that the increased stratification increases the unresolved turbulent shear, and does not directly enhance mixing. Danabasoglu et al. (2006) define N osbl as N osbl = b(k osbl+1/2) b(k osbl 1/2 ) z(k osbl+1/2 ) z(k osbl 1/2 ), (53) where k osbl is the index for the grid cell that is closest to the OSBL depth (see Figure 3). If equation (53) is used in equation (52), there may be circumstances when the stratification is too weak. To illustrate, consider two snapshots of N 2 from the low-resolution simulation, which are shown in Figure 4d. These profiles are from the times indicated by the arrows in Figure 4a. During rapid deepening (t 1 ), the value of N osbl in equation (52) should be that at the model level nearest the OSBL depth, as it is the maximum value near the OSBL base. When the boundary layer is not deepening as rapidly (t 2 ), N osbl should be one level higher in the water column. Thus we could instead define N osbl as N osbl = b(k osbl 1/2) b(k osbl 3/2 ) z(k osbl 1/2 ) z(k osbl 3/2 ). (54) 37

38 (a) (b) (c) (d) Figure 4: Test Case Convect I: Results from the baseline configuration (as defined in Table 3) for uniform vertical grid spacings of dz=1m (red lines) and dz=10m (blue lines). The LES output is dashed black and the analytic solution (see (Appendix D)) is solid black. (a) Boundary layer depths computed using equation (22). The analytic solution is from equation (D.10). (b) Normalized turbulent buoyancy fluxes. The KPP turbulent fluxes are computed via the parameterizations in equations (3), (4), and (24). (c) Buoyancy profiles averaged over a three hour window centered at the beginning of day 6. (d) Snapshots of Brunt Väisälä frequency (N 2 ; units of 10 5 s 2 ) at two separate times indicated by arrows in (a) to illustrate the need to use equation (55). The blue line is from time t 1 and the red line from time t 2. Note that the KPP results from the three models using dz=1 m are indistinguishable. Also note that LES results in panels (b) and (c) were used to construct Figure 2. 38

39 Even though use of equation (54) in place of (53) will increase OSBL depths at high resolution, it is a fragile, if not dangerous, solution. If equation (54) is used at coarse resolution, there is a possibility that N osbl 0. Thus we propose ( b(kosbl+1/2 ) b(k osbl 1/2 ) N osbl = max z(k osbl+1/2 ) z(k osbl 1/2 ), b(k ) osbl 1/2) b(k osbl 3/2 ). (55) z(k osbl 1/2 ) z(k osbl 3/2 ) OSBL depths from a test using equation (55) in the Vt 2 are shown in Figure 5. With the new definition of N osbl, the shallow bias at high resolution is greatly improved. The coarse resolution result is nearly unchanged from the Base case (Figure 4(a)) as the entrainment layer is strongly smoothed and the stratification is constant below the mixed layer. The high frequency OSBL depth noise in Figure 4a is gone in Figure 5. The increased OSBL entrainment overcomes the artificially induced stratification from the non-local parameterization, which is discussed further in Section Sensitivity to Model Configuration In certain configurations, KPP can become unstable. For example, when enhanced diffusivity is disabled, which is consistent with disabling of internal matching, large temporal noise develops in the OSBL depths (Figure 6a). These large oscillations are evident similar MOM6 and POP configurations as well (not shown). Time integrated profiles of buoyancy and buoyancy flux (Figures 6b and c respectively) suggest that the impact of the large OSBL oscillations is minimal. Yet, the rapidly oscillating OSBL depths could have strong interactions with other parameterizations (e.g., symmetric instability, Bachman et al., 2017). The oscillations do not appear at high resolution (recall enhanced diffusivity is minimal at high resolution; Section 4.5). In all cases without the enhanced diffusivity parameterization, the negative buoyancy flux minimum is shallower than the baseline configuration (Figure 6b). The characteristics of the OSBL noise is sensitive to how CVMix interfaces with the calling model. Possible interface configurations are illustrated in Figure 7. Originally MPAS-O used Option (2) in Figure 7, where the non-local tendency (equation (24)) and surface flux tendencies are computed and applied before calling CVMix. Static stability is then diagnosed and CVMix is called. The resulting OSBL depths from this configuration are 39

40 Figure 5: Test Case Convect I: Boundary layer depths computed using equation (22) where the analytic solution is from equation (Appendix D). These simulations are similar to the baseline configuration (Table 3) except N osbl in equation (33) follows equation (55). The baseline results from MPAS-O are plotted as dashed lines. 40

41 MPAS NM 10m in Figure 6a. If the surface and non-local flux tendencies are computed and applied after calling CVMix, which is the physically consistent sequence (Option (3) in Figure 7), the magnitude and duration of the noise is slightly reduced (MPAS NM Order1). MPAS-O now computes and applies the surface flux tendency before CVMix and the non-local flux tendency is applied after (Option (1) in Figure 7), the noise is greatly reduced (MPAS NM Order2). This is due to the decrease in surface layer buoyancy via surface cooling in this test case. The decreased surface layer buoyancy will increase the OSBL depth (equation (22)), which increases entrainment, minimizing the noise. Finally, if we use equation (55) in the parameterization for Vt 2, the noise is gone. This is again due to the increased entrainment observed in Figure 5 and illustrated in Figure 4d. The root cause of this noise is the artificial warming associated with the cubic shape function chosen by LMD94 for non-local transport. The non-local flux parameterization, which along with the explicit model time stepping, causes artificial warming near the surface (assuming θ/ z 0), which shallows the OSBL. Convection associated with the surface cooling quickly homogenizes the near surface warming and deepens the OSBL. To illustrate the influence of the cubic shape function in the non-local flux parameterization (equation (24)), the PNL test (Table 4) uses a parabolic non-local shape function (equation (43)) in the non-local flux parameterization (equation (24)). Figure 8 shows how changing the non-local shape function mitigates the OSBL noise, but at the cost of an increased shallow bias at coarse resolution. Recall that equation (43) is monotonic in the OSBL and thus is not prone to the near surface artificial warming (Section 4.4.5) and associated noise. It is also possible that the consistency with LES and baseline KPP in Figure 8 is due to increased entrainment near the OSBL base (the slope of equation (43) is shallower at σ = 1 relative to cubic of LMD94). Further the increased entrainment from increased Vt 2 mitigated similar highfrequency noise seen near the beginning of the run at high resolution in the baseline configuration Recommendations The sensitivity tests in this section suggest that to mitigate possible KPP biases and instabilities, models must use LMD94 enhanced diffusivity, or 41

42 (a) (b) (c) Figure 6: Test Case Convect I: OSBL depths from KPP simulations where enhanced diffusivity has been disabled to be consistent with no matching of interior diffusivities. In MPAS-O, the surface flux and non-local flux tendencies are computed and applied prior to calling CVMix (Option (2) in Figure 7). In MPAS NM Order1 the surface flux and non-local flux tendencies are computed and applied after the call to CVMix and before the implicit vertical mixing solver (consistent with LMD94; Option 3 in Figure 7). For MPAS NM Order2, the surface flux tendency is computed and applied before CVMix and the non-local flux tendency is applied after (Option (1) in Figure 7). MPAS NM N2 computes tendencies as in MPAS NM Order1, except N in equation (33) is taken as max(n Kosbl, N Kosbl+1 ) (equation (55)). The LES output is dashed black and the analytic solution is solid black. The baseline configuration results are reproduced as dashed lines. (a) Boundary layer depths computed using equation (22) where the analytic solution is from equation (D.10). (b) time averaged buoyancy profiles and (c) time averaged, normalized buoyancy fluxes, where KPP buoyancy fluxes are computed via the parameterizations in equations (3), (4), and (24). 42

43 (1) (2) (3) Figure 7: Schematic illustrating the three different interfaces between MPAS-O and CVMix. The top box shows the predictive equation for Convect I. In option (1), the surface fluxes are computed and applied. Next the updated potential temperature profile (θ S ) is sent to CVMix to compute diffusivity profiles. The local and non-local fluxes are then applied (bottom equation). In option (2), the non-local and surface flux tendencies are computed and applied yielding an intermediate potential temperature (θ NS ). Diffusivity profiles (κ NS ) are computed based on θ NS. The local mixing is then computed using κ NS. Option (3), which is most consistent with LMD94 computes diffusivities based on time level n potential temperature values and then all terms are computed and applied. 43

44 Figure 8: Test Case Convect I: As in Figure 6a, but for test PNL, where the parabolic profile shown in equation (43) has been used in the non-local flux parameterization (equation (24)). The dashed color lines are the MPAS-O baseline configuration results. 44

45 a shape function for non-local transport that is monotonic in the OBL, or redefine the value of N (equation (55)) in the Vt 2 (equation (33)). parameterization Of these choices, we strongly recommend use of either LMD94 enhanced diffusivity (this is consistent with the baseline configuration) or the alternate shape function for non-local transport (Figure 8). The latter also reduces resolution bias seen in the baseline configuration (compare Figure 4a to Figure 8), but this improvement results in increased OSBL depth bias relative to LES. The high-resolution shallow bias seen in the baseline configuration can also be reduced by using enhanced diffusivity and changing the definition of N in the Vt 2 parameterization (Figure 5) Convect II In this test, which uses MPAS-O data, temperature and salinity are uniform in the upper 25 m. At 25 m depth, the salinity increases by 1 ppt in one model layer and the temperature decreases linearly below 25 m. We impose a surface sensible heat flux of 75 W m 2 (cooling). There is no surface forcing of salinity or momentum. This test very roughly recreates a case where fresh river water sets up a halocline and a shallower thermocline that is eroded by surface cooling. Given there are no evaporative fluxes and near surface salinity gradients in Convect II, KPP will predict a near zero salinity flux in the OSBL (see equations (3) and (24)). In contrast, LES of Convect II predicts a large salinity flux (Figure 9c). Analysis of LES results (Section 6.2.2) will show that this is due to the KPP assumption that the non-local flux parameterization simply redistributes the surface flux. LES results show a large non-local flux despite the lack of a surface flux. The assumptions exercised in Convect II are common to all tested configurations of KPP (Table 4) thus we focus on the high resolution baseline configuration alone. Results show (Figure 9) that the assumed non-local parameterization in KPP (equation (24)) can lead to important pathologies and large temperature and salinity biases. 45

46 Basic behavior for Convect II Figure 9 shows the evolution of salinity, temperature, and normalized salinity flux for LES and salinity and temperature biases from MPAS-O KPP output relative to LES. Very quickly the LES salinity profiles (Figure 9a) mix across the initial salinity jump (z = 25m) leading to large KPP biases (Figure 9d). The near surface salinity in LES increases relative to KPP, while salinity below z = 25m decreases (compare Figures 9a and d). The salinity mixing creates two regions of strong stratification, the first between the near surface well mixed layer, and the second at depth below the region of active mixing (near z = 34m by day 4). The LES OSBL depth, diagnosed with the KPP algorithm, places the boundary layer at the shallower depth (dashed line in Figure 9). Consistent with a shallower OSBL, the LES temperature cools near the surface (Figure 9b). This creates a warm bias in KPP (Figure 9e) near the surface, where mixing is strongest. Slightly shallower than z = 25m, KPP is colder than LES and slightly deeper, LES is colder. These results suggest that KPP is missing a source of entrainment seen in the LES results. The normalized salinity flux (Figure 9c) shows that turbulence exists below the OSBL depth diagnosed using the KPP algorithm. Slightly below z = 25m, a strong salinity flux remains, similar in magnitude to the near surface flux. This flux at depth exists below the KPP diagnosed OSBL depth further suggesting the difficulty the KPP algorithm will have to simulate this case. Even if the critical Richardson number was altered to deepen the boundary layer below the LES salinity fluxes, the KPP algorithm could not reproduce the bimodal flux profile in Figure 9c Non-local transport in the absence of surface fluxes To determine the source of the salinity flux in LES, which is missing in KPP, consider an approximate equation for the vertical turbulent salinity flux, which following Section 4.4 can be written as w S τw 2 S z }{{} local gβ S S 2 1 w w S } w {{ 2 } w 2 z. (56) }{{} buoyant production triple moment In equation (56) we interpret the salinity flux as in KPP, such that the nonlocal transport is seen as a correction when the local gradient of salinity is 46

47 (a) (d) (b) (e) (c) Figure 9: Test Case Convect II: Results from the Convect II test case, showing (a) salinity (ppt) from LES, (b) potential temperature ( C) from LES, (c) normalized salinity flux from LES (d) salinity (ppt) and (e) potential temperature ( C) biases. The bias is defined as MPAS-O baseline KPP configuration results (Table 3) at dz = 1m resolution relative to LES (MPAS-O - LES). The dashed line is the LES OSBL depth. The OSBL depth is diagnosed using equation (22). Note in this case KPP predicts no salinity flux. 47

48 small. In KPP, the triple moment term is assumed small, and the product of τ w 2 and the buoyant production term is interpreted, and parameterized, as the non-local redistribution of the surface salinity flux. In Convect II, there is no surface flux of water so the buoyant production term is zero in KPP. Figure 10 shows the vertical profiles of the three terms in equation (56) (normalized by τw 2 ) from LES averaged over a three hours period centered at 1.5 days. The triple moment term is indeed small. The local contribution to the salinity flux is also small due to the small salinity gradient (Figure 9a) in the upper ocean and lack of vertical velocity variance at depth. The buoyant production term, on the other hand is large in LES. Thus a strong non-local redistribution of salinity is present that KPP does not represent. This leads to the large biases evident in Figure 9. The biases evident here would also extend to passive tracers in the ocean that do not experience a surface flux. For example, Nicholson et al. (2016) suggest that iron supplies in the boundary layer of the Southern Ocean are strongly influenced by entrainment, which is enhanced by passing storms. It seems likely that the storms are not accompanied by a surface flux of iron and thus KPP will not be able to correctly simulate biomass of phytoplankton under these conditions Recommendations No solutions to the biases seen in Figure 9 are readily apparent. Accurately simulating Convect II requires an extension of the KPP non-local flux parameterization beyond the redistribution of surface fluxes. One possible solution is to follow Noh et al. (2003), who proposed a non-local flux parameterization that redistributes a boundary layer entrainment buoyancy flux in addition to the surface buoyancy flux. A parameterization of this form may address some of the issues seen in Convect II Convect III Convect III is similar to Convect I, but includes a small evaporative flux (1.37 mm/day) and a moderate wind stress (τ = 0.1 Pa). In this test the shear instability scheme from LMD94 is used. In the baseline configuration the shear instability scheme influences the OSBL via matching to KPP predicted diffusivities and viscosities at the OSBL (equations (18a) - (18b)). The matching technique proposed by LMD94 assumes that the OSBL and the interior ocean follow different mixing rules, which is consistent with some atmospheric literature (e.g., Wyngaard, 1983). Yet, many boundary layer 48

49 x10-7 Figure 10: Test Case Convect II: Temporally averaged (three hours) salinity flux (m ppt s 2 ) budget terms predicted by LES (as defined in equation (56)) at t=1.5 days. Individual terms are given in the legend. All terms have been normalized by τw 2, yielding units of ppt m 1. Local production is proportional to the gradient of salinity, buoyant production is proportional to the salinity variance, and the triple moment term is the vertical divergence of the turbulent transport of the turbulent salinity flux. 49

50 parameterizations (e.g., Mellor and Yamada, 1974) do not require a clean separation between internal and boundary layer mixing. This test examines the sensitivity of KPP to matching diffusivities and viscosities predicted by interior mixing schemes at the OSBL base. Results show that if interior diffusivities are matched to KPP predicted diffusivities, linear interpolation must be used to compute ν int (h) in equations (18a) - (18b) (Figure 14). If KPP diffusivities are not matched to the LMD94 shear instability scheme, the interior diffusivities must be extended into the OSBL (Figure 12. The extension of internal mixing into the boundary layer inherently assumes that KPP parameterizes eddies of a different temporal and spatial scale than internal mixing. Both configurations provide decent agreement with LES OSBL depths (Figures 11 and 12) Baseline Results Results from the baseline configuration of Convect III (Table 3) are shown in Figure 11 for MPAS-O, MOM, and POP. As in Convect I, all models have very similar solutions. Unlike Convect I, there is no shallow OSBL bias at high resolution as the model resolved shear dominates over the influence of the Vt 2 parameterization in the KPP OSBL depth prediction (equation (22)). Thus the resolution dependence seen in Convect I is minimized. Vertical profiles of zonal momentum (Figure 11c) suggest that KPP does not mix as effectively as LES. Between depths of meters, LES zonal momentum profiles are more uniform in the vertical than KPP. A similar result is seen in meridional momentum profiles (Figure 11d). The presence of momentum fluxes in regions of small local momentum gradient suggests a role for non-local momentum transport. A parameterization for non-local momentum transport has been proposed (Smyth et al., 2002), but has not been sufficiently verified to recommend inclusion in CVMix Influence of matching to interior schemes OSBL depths from tests disabling the matching of KPP predicted diffusivities and viscosities to those predicted by the LMD94 shear instability scheme are shown in Figure 12. When the LMD94 shear instability scheme is constrained to the interior only (in tests with no diffusivity matching), simulated OSBL depths are too shallow at high resolution (MPAS NM noadd 1m in Figure 12). The shallow OSBL bias is due to the inability of shear instability mixing to influence the OSBL (consistent with LMD94). When the shear instability scheme is extended into the OSBL, the simulated OSBL 50

51 Figure 11: Test Case Convect III: (a) OSBL depths (m) calculated using equation (22), (b) buoyancy, (c) momentum in the wind stress direction, and (d) momentum in the direction perpendicular to the wind stress direction. All profiles are averaged over an inertial cycle on day seven. In each panel, MPAS-O results are solid, MOM are dashed, and POP are dot-dashed. High resolution results are red and coarse are blue. All models follow the baseline Convect III configuration (Table 3). 51

52 depths deepen and are consistent with the baseline configuration (compare Figures 12 to 11a). At coarse resolution, the inclusion of shear instability mixing in the OSBL does not dramatically alter the results (not shown). It is likely that the insensitivity to interior mixing in the OSBL is due to the weakened current shear and buoyancy gradients resulting from large dz Entrainment at the OSBL base When the LMD94 enhanced diffusivity parameterization is disabled, the OSBL shallows relative to the baseline configuration (Figure 12), which is consistent with Convect I (Figure 6). Unlike Convect I, high-frequency, hightemporal, noise in the OSBL depths is not seen (compare Figure 12 to Figure 6). It is likely that mean shear increases entrainment via its dominance over the Vt 2 parameterization and via the LMD94 shear instability scheme, which mitigates the high-frequency noise in boundary layer depths. This is similar to the decrease in noise in Convect I when LMD94 enhanced diffusivity is enabled. If matching is disabled and a monotonic shape function (equation (43)) is used for the non-local tracer transport parameterization, the OSBL depths (Figure 13) are similar to the baseline configuration (Figure 11a). Thus LMD94 enhanced diffusivity is not required if an alternate shape function is used in the non-local parameterization. The coarse resolution is slightly improved with the parabolic shape function for non-local transport relative to the baseline configuration. Despite not improving OSBL biases relative to the baseline configuration, the no-match (NM test in Table 4) configuration does not require the complexities associated with matching viscosities and diffusivities between the KPP scheme and internal mixing parameterizations Diffusivity matching techniques The baseline configuration interpolates diffusivities and viscosities to the OSBL depth using a weighted linear interpolation (Appendix D of LMD94). Previous analysis of KPP (e.g., Large and Gent, 1999; Danabasoglu et al., 2006) have only considered linear interpolation of diffusivities and viscosities. If interior mixing varies rapidly near the OSBL base, a higher order interpolation may be appropriate. Sensitivity to the interpolation method is shown in Figure 14. The OSBL bias is relatively unchanged at coarse resolution. Yet at high resolution, the OSBL depths vary rapidly in time (red 52

53 Figure 12: Test Case Convect III: OSBL depths calculated with equation (22) from MPAS-O sensitivity tests examining the influence of not matching to interior mixing. MPAS NM noadd uses equation (20) as the shape function in equation (4). MPAS NM follows MPAS NM noadd but computes shear instability mixing over the full depth. MPAS NM E follows MPAS NM but disables enhanced diffusivity. The results from the MPAS-O baseline configuration are shown as blue and red dashed lines. 53

54 Figure 13: Test Case Convect III: OSBL depths (calculated from equation (22)) from MPAS-O sensitivity tests examining the influence of shape function choice in the nonlocal flux parameterization (equation (24)). MPAS PNL utilizes equation (43) in place of equation (20) in the non-local flux parameterization (equation (24)). The results from the MPAS-O baseline configuration are shown as blue and red dashed lines. 54

55 Figure 14: Test Case Convect III: OSBL depths (m) with time. MPAS Match Q follows the baseline configuration but uses quadratic interpolation to compute diffusivities at the OSBL depth, MPAS MatchLimit Q follows MPAS Match Q, but forces interpolated diffusivities and gradients to be positive. The results from the MPAS-O baseline configuration are shown as blue and red dashed lines. 55

56 line in Figure 14). Quadratic interpolation also leads to unphysical behavior buoyancy and momentum profiles (not shown). The unphysical behavior in this test case results from the lack of monotonicity in quadratic interpolation. When internal diffusivities change sharply near the OSBL base, quadratic interpolation can yield negative diffusivities. If an artificial limit is placed on the interpolated OSBL diffusivity, the unphysical behavior is mitigated (no shown), however, the results are not significantly improved relative to the baseline configuration. Therefore, if matching is used in KPP, linear interpolation is strongly recommended. In the baseline configuration of KPP, the diffusivity and gradient of diffusivity are matched to that predicted by internal mixing parameterizations (equations (18a) - (18b)). Given potential difficulties with representing diffusivity gradients, we test the influence of matching to OSBL diffusivities alone (Match Diff in Figure 15). At high resolution there is little change in simulated OSBL depths (Figure 15a), but there is an increased OSBL shallow bias for dz = 10 m. The time-averaged buoyancy profile for the Match Diff test (Figure 15b) is under-mixed relative to the other profiles and the baseline configurations (Figure 11b). Limiting of diffusivities is not required for linear interpolation as it is already constrained positive. To alleviate potential issues in matching to diffusivity gradients, the interior gradient should take the same sign as the KPP diffusivity gradient at the OSBL (positive). When the diffusivity gradient is constrained to be positive, the result is similar to the baseline configuration (compare MatchLimit test to baseline configuration in Figure 15 ). In more realistic configurations, other diffusivity parameterizations (e.g. tidal mixing) could result in much larger diffusivity gradients and limiting may be necessary Non-Local momentum fluxes Throughout most of the column, the zonal momentum profile from KPP is fairly consistent with LES (Figure 11c). At mid-depths the LES profile of zonal and meridional momentum is well mixed, whereas KPP is not, suggesting an important role for non-local momentum transport. It is possible that the shallow OSBL bias across all KPP tests relative to LES (including tests with minimal bias in Convect I, e.g. MPAS N2 1m) in Convect III is due to the lack of non-local momentum transport. 56

57 Figure 15: Test Case Convect III: (a) OSBL depths (m), (b) buoyancy profiles, (c) momentum profiles in the wind stress direction, and (d) momentum profiles in the direction perpendicular to the wind stress direction. All profiles are averaged over day seven (a full inertial cycle). MPAS Match Diff follows the baseline configuration but matches interior diffusivities but not gradients and MPAS MatchLimit matches interior diffusivities and gradients but constrains both to be positive. 57

58 Recommendations The results of Convect III suggest two viable configurations of KPP: If the baseline configuration is used, we recommend using linear interpolation as in LMD94 (their Appendix D) to compute interior diffusivities/viscosities at the OSBL base (Figure 14), and matching should be to both the internal diffusivity and its gradient (Figure 15). If the no-match configuration configuration is used, we recommend keeping the LMD94 enhanced diffusivity or use the parabolic shape function in the non-local parameterization (compare Figure 13 to MPAS NM E 10m in Figure 12). extending interior mixing parameterizations into the OSBL (Figure 12), and 6.4. Diurnal I In addition to destabilizing surface cooling, a diurnal cycle is imposed in Diurnal I. There is no net heat input as the daily averaged buoyancy loss from cooling and evaporation is exactly balanced by daily averaged buoyancy input by solar radiation. The resolution dependent OSBL bias seen in Convect I is also evident in this test (Figure 16a). The alteration to N (equation (55)) in the Vt 2 parameterization again diminishes this resolution dependent bias (Figure 16b). At high resolution, high frequency noise develops in OSBL depths during the period of stabilizing buoyancy forcing (Figure 16a). This noise is reduced when the monotonic (parabolic) shape function (equation (43)) is utilized in the non-local transport parameterization (Figure 18). We have also tested the sensitivity to the amount of shortwave radiation carried by the non-local heat flux. The amount varied from the shortwave radiation absorbed in the top layer to that absorbed in the entire OSBL. Across these tests, we found very little variation in the simulated OSBLs (not shown). This is consistent with LMD94 (see their Figure C4, Appendix C) as this study (and LMD94) use an equation of state where density increases until freezing occurs (e.g., equation (48)). Near the coast (low salinity) the maximum density occurs above freezing, and shortwave radiation 58

59 can be destabilizing. This could lead to extreme sensitivity to the amount of shortwave radiation carried in the non-local temperature flux Resolution Dependence In Diurnal I we also consider a third vertical discretization: a non-uniform grid with 1.5-m resolution near the surface. Figure 16a shows the OSBL depths for the baseline configuration (Table 4) of Diurnal I for the three resolutions. At highest resolution, there is a shallow bias during destabilizing surface buoyancy forcing. As resolution coarsens, the OSBL bias improves (non-uniform grid, tagged S in Figure 16) and then becomes a slight deep bias at dz = 10 m. At the two higher resolutions, high frequency temporal OSBL noise develops near the surface boundary during stabilizing buoyancy forcing (Figure 16a). This is similar to what was seen in Convect I (Figure 6, near the start of the simulation). Yet, unlike Convect I, in Diurnal I the noise develops even with the LMD94 enhanced diffusivity parameterization enabled. This suggests that including the enhanced diffusivity parameterization to mitigate noise is not robust for regions with strong diurnal forcing. When the chosen model level of N in the Vt 2 parameterization is the stratification within the entrainment layer (equation (55)), the resolution dependence of the OSBL bias is slightly reduced (Figure 16b). But the near-surface noise remains. The noise also appears correlated to a delay in transition from shallow to deep boundary layers (compare red and green to blue OSBL deepening in Figure 16a). Figure 17 shows the temperature bias (MPAS-O - LES) for the three resolutions (top - dz = 1m, middle - dz = 10m, bottom = stretched grid). The baseline configuration (left column) shows a strong resolution dependence in the temperature bias. The non-uniform grid has the least bias, but it grows through the course of the simulation. This bias growth is also evident in the high-resolution result. When the chosen model level of stratification is changed, the bias in the high resolution and non-uniform grid cases is reduced. The coarse resolution case remains relatively consistent with the baseline configuration. Given that N(k osbl 1 ) N(k osbl ) at coarse resolution, the consistent behavior at dz = 10m is expected OSBL noise In Convect I, it was suggested that the artificial warming induced by the non-local flux parameterization could create near surface stable stratification 59

60 (a) (b) Figure 16: Test Case Diurnal I: OSBL depths (m), here S indicates use of a stretch grid, top layer dz = 1.5m. OSBL depths from (a) the baseline configuration (Table 3), (b) test examining the influence of changing the model level of N in the unresolved turbulence parameterization (equation (33)). 60

61 (a) (d) (b) (a) (e) (c) (f) Figure 17: Test Case Diurnal I: Temperature (MPAS-O - LES) bias ( C) for MPAS Base (a), (b), and (c) and MPAS N2 (d), (e), and (f). (a) and (d) are dz = 1m, (b) and (e) are dz = 10m, and (c) and (f) use a stretched grid with a top layer of 1.5m. In all plots, the solid black line is the KPP predicted OSBL depth and the dashed black line is the LES OSBL depth. All OSBL depths are computed using equation (22). 61

62 that led to an unphysical shallowing of the OSBL. The non-physical stratification near the surface could contribute to the high frequency noise seen in Figure 6. Unlike Convect I, the noise in Diurnal I is most evident for shallow boundary layers and at high resolution. At high resolution, the influence of the LMD94 enhanced diffusivity parameterization is minimal. Further, when the OSBL is shallow ( 5m) there are only a few model layers in the OSBL. Thus, the noise seen in the high resolution (Figure 16a) case is roughly equivalent to the low resolution case in Convect I (Figure 6; MPAS NM 10m). When there are only a few layers in the surface layer, the cubic shape function results in stronger near-surface artificial stratification and hence OSBL depth noise. Finally, recall that the noise in the high resolution Diurnal I simulation is also evident in the high resolution Convect I simulation (Figure 4a, near the beginning of the simulation). When the definition of N in the Vt 2 parameterization is altered (equation (55)) the noise for shallow OSBLs is reduced, yet remains (Figure 16b), which is consistent with Convect I (Figure 5). If the parabolic shape function (equation (43)) is used in place of the original cubic function for the non-local flux, the OSBL depth noise is removed (Figure 18), which is consistent with the Convect I PNL test (Figure 8). Note also, that the simulated boundary layers track the LES solution more faithfully than the baseline configuration, especially in the transitions between shallow and deep boundary layers. In Figure 18, the resolution dependent bias has reemerged, but to a lesser degree. If we define the value of N in the Vt 2 parameterization as the maximum of N(k osbl 1 ) and N(k osbl ) (equation (55)) and use the parabolic shape function in the non-local transport parameterization, the maximum OSBL depth at high resolution becomes similar to the coarse-resolution result (Figure 18) Recommendations From the results of Diurnal I, we can recommend the following: Defining N as the maximum of N(k osbl 1 ) and N(k osbl ) (equation (55)) in the Vt 2 parameterization (equation (33)) can be used to mitigate most of the resolution dependent bias seen in Figure 16a, or Using a monotonic shape function (equation (43)), which mitigates the noise seen for shallow OSBL depths at high resolution. 62

63 Figure 18: Test Case Diurnal I: OSBL depths (m), here S indicates use of a stretch grid, top layer dz = 1m. OSBL depths examining the influence of using an alternate shape function for the non-local flux (equation (43) used in equation (24)) An additional test (PNL N2) that changes the model level of N (equation (55)) in the unresolved turbulence parameterization (equation (33)) and the non-local flux shape function. 63

64 Diurnal II Diurnal II follows from Diurnal I, except a zonal wind stress of 0.1 Pa and a small evaporative flux are included. Thus this test case is not truly in equilibrium, but results show that the rate of change in OSBL depth is relatively small over the last few days of the simulation (e.g., Figure 19a). As in Convect III, the LMD94 shear instability scheme and LMD94 matching is enabled in the baseline configuration. The results from Diurnal II (Figure 19) suggest that KPP transports too much heat toward the OSBL base during stabilizing buoyancy forcing. This leads to enhanced stratification across the OSBL base, which can artificially limit OSBL deepening Baseline configuration In the baseline configuration, KPP is too efficient at the downward transport of heat during stabilizing surface buoyancy forcing (see anomalously strong negative buoyancy flux bias in Figure 19b-c). While the maximum buoyancy flux is roughly equivalent to LES, the transition from convective to stable conditions is too rapid. This is why biases in Figure 19b-c appear similar to Figure 19a but a slightly shifted temporally and weaker in magnitude. KPP predicts vigorous buoyancy fluxes above the diagnosed OSBL. This is not the case in LES (Figure 19a), where buoyancy fluxes become negligible 5-10 meters above the OSBL. This anomalously strong, stabilizing, buoyancy flux creates artificially strong stratification near the OSBL base. This can lead to artificial deepening of the OSBL. Recall that the unresolved turbulent shear, which is important during portions of the inertial cycle when the current opposes the wind stress (Noh et al., 2016), is directly related to the stratification in the entraining layer. Increased values of Vt 2 decreases the bulk Richardson number, which deepens the OSBL (equation (22)). The slight deep bias in the KPP OSBL depths is not seen at coarse resolution due to the inability to simulate strong stratifications near the OSBL (Figure 19c) Non-local tracer flux Similar to Convect III, there is no high-frequency noise in the OSBL depths. However, the chosen shape function for non-local transport is important. The normalized buoyancy fluxes and OSBL depths from a MPAS-O test using equation (43) in the non-local transport scheme of KPP (equation (24)) are shown in Figure 20. The anomalously strong buoyancy flux 64

65 (a) (b) (c) Figure 19: Test Case Diurnal II: Normalized buoyancy fluxes ( w b /w b sfc) through time for LES (a), and the MPAS-O baseline configuration bias relative to LES (MPAS-O - LES) with dz = 1m (b), and dz = 10m (c). In every panel the solid black line is the OSBL depth computed via equation (22) for each simulation. In (b) - (c) the LES OSBL depth is shown as the solid black line for comparison. 65

66 (a) (b) Figure 20: Test Case Diurnal II: As in Figures 19b and c but using the parabolic shape function in the non-local flux parameterization. 66