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A Generalized Slab Model

Ian A. Stokes aScripps Institution of Oceanography, University of California, San Diego, La Jolla, California
bDepartment of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, California

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https://orcid.org/0000-0002-3818-8710
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Samuel M. Kelly cLarge Lakes Observatory, University of Minnesota Duluth, Duluth, Minnesota
dDepartment of Physics, University of Minnesota Duluth, Duluth, Minnesota

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Andrew J. Lucas aScripps Institution of Oceanography, University of California, San Diego, La Jolla, California
bDepartment of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, California

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Amy F. Waterhouse aScripps Institution of Oceanography, University of California, San Diego, La Jolla, California

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Caitlin B. Whalen eApplied Physics Laboratory, University of Washington, Seattle, Washington

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Thilo Klenz fCollege of Fisheries and Ocean Sciences, University of Alaska Fairbanks, Fairbanks, Alaska

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Verena Hormann aScripps Institution of Oceanography, University of California, San Diego, La Jolla, California

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Luca Centurioni aScripps Institution of Oceanography, University of California, San Diego, La Jolla, California

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Online Publication:
13 Mar 2024
Print Publication:
01 Mar 2024
DOI:
https://doi.org/10.1175/JPO-D-23-0167.1
Page(s):
949–965
Open access
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Abstract

We construct a generalized slab model to calculate the ocean’s linear response to an arbitrary, depth-variable forcing stress profile. To introduce a first-order improvement to the linear stress profile of the traditional slab model, a nonlinear stress profile, which allows momentum to penetrate into the transition layer (TL), is used [denoted mixed layer/transition layer (MLTL) stress profile]. The MLTL stress profile induces a twofold reduction in power input to inertial motions relative to the traditional slab approximation. The primary reduction arises as the TL allows momentum to be deposited over a greater depth range, reducing surface currents. The secondary reduction results from the production of turbulent kinetic energy (TKE) beneath the mixed layer (ML) related to interactions between shear stress and velocity shear. Direct comparison between observations in the Iceland Basin, the traditional slab model, the generalized slab model with the MLTL stress profile, and the Price–Weller–Pinkel (PWP) model suggest that the generalized slab model offers improved performance over a traditional slab model. In the Iceland Basin, modeled TKE production in the TL is consistent with observations of turbulent dissipation. Extension to global results via analysis of Argo profiling float data suggests that on the global, annual mean, ∼30% of the total power input to near-inertial motions is allocated to TKE production. We apply this result to the latest global, annual-mean estimates for near-inertial power input (0.27 TW) to estimate that 0.08 ± 0.01 TW of the total near-inertial power input are diverted to TKE production.

© 2024 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Ian A. Stokes, istokes@ucsd.edu

Abstract

We construct a generalized slab model to calculate the ocean’s linear response to an arbitrary, depth-variable forcing stress profile. To introduce a first-order improvement to the linear stress profile of the traditional slab model, a nonlinear stress profile, which allows momentum to penetrate into the transition layer (TL), is used [denoted mixed layer/transition layer (MLTL) stress profile]. The MLTL stress profile induces a twofold reduction in power input to inertial motions relative to the traditional slab approximation. The primary reduction arises as the TL allows momentum to be deposited over a greater depth range, reducing surface currents. The secondary reduction results from the production of turbulent kinetic energy (TKE) beneath the mixed layer (ML) related to interactions between shear stress and velocity shear. Direct comparison between observations in the Iceland Basin, the traditional slab model, the generalized slab model with the MLTL stress profile, and the Price–Weller–Pinkel (PWP) model suggest that the generalized slab model offers improved performance over a traditional slab model. In the Iceland Basin, modeled TKE production in the TL is consistent with observations of turbulent dissipation. Extension to global results via analysis of Argo profiling float data suggests that on the global, annual mean, ∼30% of the total power input to near-inertial motions is allocated to TKE production. We apply this result to the latest global, annual-mean estimates for near-inertial power input (0.27 TW) to estimate that 0.08 ± 0.01 TW of the total near-inertial power input are diverted to TKE production.

© 2024 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Ian A. Stokes, istokes@ucsd.edu
Keywords: Internal waves; Wind stress; Boundary layer; Oceanic mixed layer; Storm environments; Model comparison

1. Introduction

The oceanic surface boundary layer (OSBL) plays a key role in energy exchange between the atmosphere and ocean (Grachev and Fairall 2001). Wind imposes shear stress on the OSBL which drives surface waves and turbulent mixing in the upper ocean. With time, a mixed layer (ML) develops that extends downward from the ocean surface. The ML is a ubiquitous feature of the OSBL, characterized by an approximately homogeneous region with small vertical gradients in temperature and salinity that exists between the ocean’s surface and the stratified interior (Brainerd and Gregg 1995). The thickness of the ML varies globally and is generally largest during the winter and in energetic regions (Holte et al. 2017). Beneath the ML, stratification increases rapidly with depth, from a value near zero in the ML to a maximum value in the interior. This region of strong stratification gradients is the transition layer (TL) and marks the base of the OSBL (D’Asaro et al. 1995; Johnston and Rudnick 2009; Kaminski et al. 2021). The TL plays the important role of modulating interactions between the ML and the interior. For example, entrainment processes which enable momentum and property exchange between the OSBL and the interior are initiated in the TL and ultimately lead to ML deepening (Grant and Belcher 2011). In the analysis that follows, we investigate how variability in the TL modulates the atmosphere’s ability to supply power to the internal wave field.

Wind stress acting on the ocean surface is estimated to supply 0.27–1.4 TW of power input to the near-inertial internal wave field globally (e.g., Wunsch 1998; Alford 2001; Watanabe and Hibiya 2002; Furuichi et al. 2008; Rimac et al. 2013; Alford 2020b). Though these numbers only represent a small fraction of the total power input by the atmosphere into the ocean (Wunsch 1998; Zippel et al. 2022), energy in the internal wave band propagates vertically, creating a pathway for energy from the atmosphere to penetrate deep into the ocean’s interior (Gill 1982; Alford et al. 2012). The path begins with near-inertial oscillations (NIOs) of the ML. These oscillations are the ocean’s local response to wind forcing, and the frequency is set by the local Coriolis frequency (Plueddemann and Farrar 2006). Zonal gradients in wind stress produce variability in the intensity of NIOs, setting up regions of convergence and divergence which resonantly pump the base of the ML at the inertial frequency. This pumping action converts the inertial oscillations into near-inertial internal waves (NIWs; Gill 1984). Meridional variability of the Coriolis effect (i.e., the β effect: D’Asaro 1989; D’Asaro et al. 1995; Moehlis and Llewellyn Smith 2001) and the submesoscale eddy field (e.g., Asselin et al. 2020; Thomas et al. 2020) set up lateral gradients in the local vorticity, which also generate NIWs. This process analogously produces convergence, but it is imposed by phase variability between adjacent NIOs rather than intensity variability (Young and Ben Jelloul 1997). Fast-moving storms (e.g., Gill 1984; D’Asaro et al. 1995; Brizuela et al. 2023) and interactions with coastlines (e.g., Pettigrew 1981; Millot and Crépon 1981; Kundu et al. 1983; Kelly 2019) can convert these oscillations into NIWs as well.

Alford and Whitmont (2007) and Silverthorne and Toole (2009) show that NIWs dominate internal wave kinetic energy and shear spectra at all depths throughout the global ocean. Observations of inertial kinetic energy (IKE) and ocean mixing show coherent seasonal variability, supporting the hypothesis that NIWs contribute strongly to ocean mixing (Alford 2020b). Thus, there is substantial evidence indicating the importance of NIWs as a means for energy from the atmosphere to reach the deep ocean. Because NIWs arise from NIOs, the potential for the generation of NIWs is set by the wind’s ability to supply power to NIOs. This power input is often referred to as the “wind work” in the literature and denoted Π (note that “wind work” is actually power input with units of W m−2). The Π sets bounds on NIW generation and is of first-order importance for energy transfer from the atmosphere to the ocean’s interior.

The slab model, introduced by Pollard and Millard (1970), is a common method for global wind work estimation (e.g., Alford 2001). The slab model operates on the simplified view that the OSBL is a homogeneous ML that responds to wind stress as a solid body. In this formulation, all of the momentum imparted by the wind is deposited in the ML, and there are no currents below. For the solid-body-ML assumption of the slab model to hold, there must exist a step-like change in stratification at the base of the ML so that the TL thickness is negligible compared to the ML thickness. This implies a linear stress profile with a step-like vertical stress gradient, mirroring the ML’s step-like stratification (Fig. 1).

Fig. 1.
Fig. 1.

A schematic is shown for an example ML, TL, and the associated stratification and stress profiles. (left) Stratification data collected from a high-resolution Argo float in the Pacific Ocean on 1 Jan 2020 at 49.4°N, 129.3°W (World Meteorological Organization Argo float ID 4902432, cycle number 57). The mixed layer depth (MLD) and the transition layer depth (TLD) are calculated from the stratification data, using the density method of Holte and Talley (2009) and the local minimum method of Sun et al. (2013), respectively. The stratification data shown have been smoothed with a 15-point moving mean. (right) The stress profiles associated with the slab model and MLTL model are plotted vs depth in black and blue, respectively. Expressions for the vertical dependence of these profiles are derived in the appendix and given by (A1) and (A2), respectively.

Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0167.1

Slab models perform well in some cases (e.g., Pollard and Millard 1970), but cannot always reproduce observations (Niiler 1975; Alford 2020b). Plueddemann and Farrar (2006) show how turbulent entrainment by inertially generated shear at the base of the ML leads to ML deepening, which cannot be resolved in a slab model. As a consequence, momentum is concentrated in the ML, leading to an overestimation of inertial currents and wind work.

During the Mixed Layer Experiment (MILE), Davis et al. (1981) observed upper-ocean shear development in response to wind stress and found that the behavior deviated considerably from the slab flow approximated by Pollard and Millard (1970). These observations showed momentum diffusion into the TL beneath the ML. D’Asaro et al. (1995) also observed nonzero inertial currents beneath the ML during the Ocean Storms experiment, which are consistent with a three-layer model consisting of a ML, a TL, and a stratified interior. D’Asaro (1995) reproduced the observed inertial currents by vertically redistributing the wind stress using a body force consistent with a linear stress profile in the ML and a quadratic stress profile in the TL. We refer to this model of the upper-ocean stress as the mixed layer/transition layer (MLTL) model.

D’Asaro’s (1995) empirical method of tuning the forcing stress (body force) successfully replicated the ML and TL currents during the Ocean Storms experiment. To generalize a tuning method, stratification based methods are needed to objectively define the ML, TL, and stratified interior. Johnston and Rudnick (2009) and Sun et al. (2013) built on D’Asaro et al.’s (1995) work by developing rigorous methods for determining the TL thickness (TLT). Johnston and Rudnick (2009) shows that the TL can be resolved by using either fine-scale shear or fine-scale stratification profiles. Sun et al. (2013) endorse the stratification approach to determining the base of the TL and define the TL as a region that extends from the base of the ML to a depth just below the maximum value of stratification. This TL depth (TLD) is selected by choosing the shallower of either a local minimum in stratification or an “abrupt change in slope” below the maximum stratification value. To avoid ambiguity associated with the latter definition, we use the local minimum method as the primary means to determining the base of the TL. If there is no local minimum in the neighborhood of the maximum, the depth of maximum stratification is used in an effort to keep the estimation of TLT conservative. A schematic for this setup that includes the stress profiles of both the slab and MLTL model with corresponding stratification is shown in Fig. 1. In this example stratification profile, the local minimum and slope methods both produce the same value for the base of the TL.

Observations show that the thickness of the TL can be equal to or greater than that of the ML (e.g., Sun et al. 2013), making the traditional slab model a poor approximation for the OSBL (see Fig. 1; D’Asaro et al. 1995; Grant and Belcher 2011). However, the impact of TLs on wind work and energetics have not been explicitly quantified. TLs substantially alter the vertical structure of near-inertial currents (D’Asaro et al. 1995; Dohan and Davis 2011). They mix momentum downward, reducing surface currents and, hence, wind work. This study develops a generalized slab model that allows for arbitrary depth-varying forcing stress profiles in the OSBL, which may be parameterized to match observations of the TL. We then analyze how the stress profile affects wind work. As a first-order improvement on the stress profile used in the traditional (ML) slab model, we add a TL using the three-layer MLTL model from D’Asaro et al. (1995).

We emphasize that the purpose of our work is to explore how variability in the vertical distribution of stress impacts the dynamics of the upper ocean’s IKE budget. The MLTL model is not intended to represent the real ocean in all cases. This model is nothing more than an idealized representation of the turbulent OSBL. However, the MLTL stress profile is a useful case study because it provides the simplest scenario where momentum imparted by the wind is allowed to penetrate beneath the ML. This stress profile specifically provides us with an estimate of how the TL affects wind work, but more generally highlights the importance of the stress profile with respect to upper-ocean dynamics.

This article is organized into three parts: theoretical development of the generalized slab model, an in situ comparison of data with the generalized slab model and the traditional slab model, and a global analysis where the large-scale significance of the TL is evaluated. In section 2, we present the physics upon which our model is based. An insight here is a proof of how nonlinear forcing stress profiles lead directly to TKE production, thus limiting the ability of wind stress to power inertial motions. We apply these ideas to the Iceland Basin (section 3a) by comparing our model with observations from NISKINe (Near-Inertial Shear and Kinetic Energy experiment). We extend the model to global analyses (section 3b) by using high-resolution Argo autonomous profiling float observations to calculate a global climatological atlas of TLT. Coupling the TLT atlas with ML climatologies of Holte et al. (2017) provides a global set of MLTL stress profiles, which allow us to quantitatively estimate how TLs impact wind work globally.

2. Methods

a. Theory

1) Governing equations

Our goal is to assess the impact of vertical stress profiles on wind work calculations using a simplified dynamical model. The linear momentum equations governing the ocean’s response to stress, ignoring pressure gradients and buoyancy forcing, are
∂u∂t+fk^×u=∂τ∂z,
where u is the velocity, f is the local Coriolis parameter, k^ is the unit normal vector, and τ is the total stress, normalized by a reference density ρ0. As we are primarily interested in inertial motions, we apply a high-pass filter to the forcing term in order to remove Ekman effects (D’Asaro 1985). The stress gradient profile provides the forcing and damping in (1) and is
∂τ∂z=∂∂z(ν∂u∂z−w′u′¯).
The second term on the right-hand side, w′u′¯, is the turbulent Reynolds stress, which dominates over the first term (the viscous stress, kinematic viscosity ν) in the ocean (Gargett 1989). The Reynolds stress depends on all of the turbulent processes in the OSBL and is difficult to measure directly (Bian et al. 2018; Huang and Qiao 2021). The stress can be modeled via direct numerical simulations (DNS) and large eddy simulations (LES) (e.g., Skyllingstad et al. 2023), but these are too computationally expensive for regional or global circulation models, which rely on simpler 1D turbulence closure models (e.g., Large et al. 1994; Umlauf and Burchard 2005). Thus, turbulence models either require high-resolution numerical grids or ad hoc parameter tuning. An alternative approach is to tune a simple conceptual model of the stress profile to fit and interpret observations (D’Asaro 1995; Plueddemann and Farrar 2006; Alford 2020b; Zippel et al. 2022). Here, we present a simple model to interpret how observed TLs alter wind work. Following D’Asaro et al. (1995), Plueddemann and Farrar (2006), and Alford (2020b), we separate stress into components due to direct wind forcing and inertial damping:
∂u∂t+fk^×u=∂τw∂z+∂τr∂z⋅ 
The first term, ∂τw/∂z, parameterizes all of the turbulent motions in the OSBL that rapidly inject wind momentum downward, such as breaking waves, Langmuir turbulence, entrainment, Stokes drift, and shear instability of the total wind-driven flow. This term may be interpreted as the body force exerted by the wind on the inertial flow (D’Asaro 1995). The second term, ∂τr/∂z, parameterizes all of the turbulent motions that specifically damp the inertial flow, which includes stratified shear instability (due to the inertial flow), bottom drag, and wave drag due to internal wave radiation (Plueddemann and Farrar 2006). In the field, these components of stress cannot be independently observed, as they are conceptual rather than objective. However, in our theoretical study this technique is useful because it allows us to separately address the forcing mechanisms from the damping mechanisms so that we may isolate the stress profile’s impact on the upper ocean’s energy budget.
We obtain an energy balance by taking (3) ⋅ u and depth integrating over the ocean depth H:
∂∂t[∫−H0|u(z)|22dz]︸IKE=[τw⋅u]0︸Πtot−∫−H0τw⋅∂u∂zdz︸PTL−∫−H0τr⋅∂u∂zdz︸PR−[τr⋅u]−H︸PB,
where we have expanded the stress-gradient terms using integration by parts, assuming τw is zero at the bottom and τr is zero at the surface. The Coriolis term, f(k^×u)⋅u=0, does no work. The kinetic energy (KE) budget indicates that the time rate of change of the inertial kinetic energy (IKE; left-hand side) is set by the balance between the total power input by wind stress (Πtot, “total wind work”) and turbulent kinetic energy (TKE) production, P. The sources of TKE production are forcing stress in the TL (PTL), internal damping stresses (PR), and bottom stress (PB). We emphasize that PTL is zero in a traditional slab model because forcing stress is confined to the ML, where shear is zero. We also note that PB can typically be neglected in the deep ocean, but may be appreciable near coastlines or otherwise shallow bathymetry.

2) TKE production in the TL

Shear production (PTL + PR) is a positive quantity that is a sink in (4) and a source term in the TKE equation (Tennekes and Lumley 1972). TKE production only occurs where the curvature in the stress profile produces velocity shear. Traditional slab models assume a linear forcing-stress profile through the ML (Pollard and Millard 1970), which produces no shear and no TKE production. Instead, energy is extracted from the flow by parameterizing the damping stress with a Rayleigh drag, ∂τr/∂z ≈ −ru, which represents the combined effects of entrainment, convective-shear instability, wave radiation, and bottom drag (Moehlis and Llewellyn Smith 2001; Plueddemann and Farrar 2006; Kelly 2019).

Price et al. (1986) used the same forcing stress profile as Pollard and Millard (1970), but substantially improved the parameterization of the damping stress. Their numerical model, Price–Weller–Pinkel (PWP), is a well-accepted 1D quasi-bulk dynamical instability ML model available for estimating vertical mixing in the TL and the atmospheric power input to inertial motions (Alford 2020a). The PWP model is widely used in large-scale global ocean models [e.g., Hybrid Coordinate Ocean Model (HYCOM); Chassignet et al. 2007]. PWP deepens the ML using a bulk Richardson number parameterization for entrainment, and accounts for TKE shear production by parameterizing stratified-shear instability of the inertial flow in the TL using the gradient Richardson number. Unstable patches are partially mixed, which smooths the velocity profile, removes kinetic energy, and transfers momentum from the ML to the TL. Plueddemann and Farrar (2006) and Alford (2020b) show that PWP agrees better with observations than traditional slab model estimates.

The generalized slab model presented here differs from the traditional slab model and PWP because it allows for an arbitrary profile of forcing stress that can extend into the TL. That is, it relaxes the traditional slab-model assumption that momentum from the wind is uniformly deposited throughout the ML. This modification is motivated by observations that the actively mixing layer, as defined by TKE dissipation, often differs from the ML, as defined by the density profile (Brainerd and Gregg 1995). With the generalized slab model, one is free to specify a forcing stress profile that was observed during a storm or modeled using LES. Moreover, one can simply specify a forcing stress profile that directly reproduces observed vertical shear in the TL (D’Asaro 1995) without having to resolve the turbulent dynamics of the OSBL.

In the analyses here, we use the MLTL forcing stress profile because it is the simplest model that produces shear in the TL (Fig. 1). The MLTL model can be combined with any model of damping stress. For computational simplicity, we use a tunable Rayleigh drag to parameterize unresolved damping stresses that depend on the strength of the inertial currents. A catch-all Rayleigh damping was also added to PWP by Plueddemann and Farrar (2006) and tuned to improve agreement with observations.

The direct generation of TKE shear production by forcing stress in the TL (PTL) is a feature of the generalized slab model that is absent in the traditional slab model. Previous models which lump TKE production into PR require that the OSBL already be in motion (u ≠ 0) in order to generate TKE. They may also underestimate PR if momentum is mixed through the OSBL by turbulence that is not directly associated with the inertial flow (e.g., wave breaking, Langmuir turbulence, etc.). In the generalized slab model, PTL is a direct function of the wind stress, so that TKE shear production occurs as soon as the wind blows, provided that the forcing stress profile has curvature. Thus, one may think of curvature in the forcing stress profile as a distinct pathway for TKE shear production in the ocean (Fig. 2).

Fig. 2.
Fig. 2.

A flowchart is provided to visualize the pathways energy can take from wind in the atmosphere to mixing. The y axis is correlated to ocean depth and is superposed on a layered model of the atmosphere–ocean system (layers in descending order are atmosphere, ML, TL, interior). Boxes indicate processes in play, and box positions indicate which layer(s) the processes act on. The vertical positions of boxes within each layer are arbitrary. Processes highlighted in red are included for completeness but are not specifically addressed in the paper.

Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0167.1

b. Solution via decomposition into normal modes

In this section we present a method to solve the governing equations derived in section 2a. In our generalized slab model, we allow for an arbitrary, depth-variable forcing stress profile, making analytical solutions in Cartesian coordinates difficult. We resolve this issue by employing the modal decomposition of Gill and Clarke (1974). For each mode, we recover the simple damped oscillator equation, analogous to Pollard and Millard (1970). Summing over all modes reconstructs the velocity profile that is generated by an arbitrary forcing stress profile. The modal approach is useful in this setting because it allows us to examine the vertical scales that are excited by the different stress profiles. In practice, if only OSBL currents are desired, a z-coordinate approach could be an effective method of solution.

1) Modal decomposition

Horizontal baroclinic velocity, u(x, z, t) = [u(x, z, t), υ(x, z, t)], is expressed as sum of orthogonal vertical modes
u(x,z,t)=∑n=1∞un(x,t)ϕn(z),
where x = [x, y], z, and t are horizontal, vertical, and time coordinates, un(x, t) is the velocity modal amplitude, n is the vertical mode number, and ϕn(z) is the vertical mode (Kelly 2016). The modes satisfy
d2Φndz2+N2cn2Φn=0 with Φn(0)=Φ(−H)=0
and
ddz(1N2dϕndz)+1cn2ϕn=0, with dϕndz|0=dϕndz|−H=0,
where ϕn = dΦn/dz, cn is the eigenspeed of mode n, and N is the buoyancy frequency. In practice (6) is solved numerically using climatological stratification. The modes are orthogonal such that
∫−H0ΦmN2cmcnΦndz=Hδmnand ∫−H0ϕmϕndz=Hδmn.
Multiplying (3) by ϕn and depth integrating yields the momentum equation for the nth mode,
∂Un∂t+fk^×Un=τnw−rUn,
where τnw is the projection of the forcing stress onto mode n and the modal transports are simply Un = Hun.

2) Forcing stress parameterization

The modal forcing stress is
τnw=∫−H0∂τw∂zϕndz.
At any given time and place, the forcing stress may be written as τw = τ0Σ(z), where τ0 is the surface wind stress and Σ(z) is the forcing profile with Σ(0) = 1. Modal forcing can then be written as
τnw=τ0ϕns with ϕns=∫−H0∂Σ∂zϕndz.
In general, both the wind stress τ0 and forcing profile Σ(z) evolve in time. Strong forcing alters the ML and TL depths through turbulent entrainment and mixing (Price et al. 1986). In the analyses here, we use static ML and TL depths based on observations and climatology. This assumption is valid for moderate winds and short-duration simulations over days to weeks. If longer simulations are desired, say weeks to months, the stress profile can be updated during forward time stepping.
When Σ(z) is constant in time and all modes are damped with the same r, the modal transports are proportional to the total transport, Un = Uϕns, where the total transport equation is
∂U∂t+fk^×U=τ0−rU.
We solve this equation numerically by specifying a wind stress time series, damping coefficient, and inertial frequency, and using a convolution method consistent with Gupta et al. (2019). We can then obtain modal amplitudes by specifying a vertical stress profile Σ(z) computing ϕns from (10), and using Un = Uϕns.

3) Wind work calculations

(i) Nomenclature

In the traditional slab model, “wind work” describes the power input by the wind to inertial motions in the OSBL. Because TKE production in the traditional slab is related to Rayleigh drag, the damping can only dissipate motions which have already been generated by the wind. Thus, the damping does not affect the “power input,” only the decay rate, so the names “wind work” and “power input” can be used interchangeably. However, in the generalized slab model, some TKE production is associated with the wind itself (PTL). This TKE production inhibits the acceleration of inertial motions, and accordingly reduces the net work on inertial motions. Specifically, some fraction of the total power input is immediately diverted to TKE production in the TL, while the remainder excites inertial motions. We refer to these quantities as the “TKE fraction” (P=PTL/Πtot) and “available wind work” (Πavail = Πtot − PTL), respectively. Since PTL is positive semidefinite, available wind work is always less than or equal to total wind work.

(ii) Total wind work
From (4), the total wind work (Πtot) is given by [τw ⋅ u]z=0 and can be expressed as
Πtot=τ0⋅usurf,
where the surface velocity can be written as a sum of modes,
usurf=1H∑n=1∞Unϕn(0)=UH∑n=1∞ϕnsϕn(0).
and the total wind work as
Πtot=τ0⋅UH∑n=1∞ϕnsϕn(0).
(iii) Available wind work
The net wind work on each mode (Πn) is given by Alford (2020a) as
Πn=τnw⋅un.
Substituting τnw=τ0ϕns and un = Uϕns/H and summing over all modes yields
Πavail=τ0⋅UH∑n=1∞ϕns2.
(iv) TKE production
The TKE production in the TL is computed by rearranging Πavail = Πtot − PTL and using the expressions above
PTL=τ0⋅UH∑n=1∞ϕns[ϕn(0)−ϕns].
(v) TKE fraction
The fraction of Πtot which is diverted to TKE production (P=PTL/Πtot) can be written using (14) and (17) so that it does not depend on wind stress
P=1−∑nϕns2∑nϕnsϕn(0).
Note that ϕn(0) only depends on observed or climatological stratification (N2) and ocean depth via (6), while ϕns additionally depends on the forcing stress profile Σ(z) via (10). This means that with stratification, ocean depth, and an inferred (or measured) stress profile, we can estimate the ratio of TKE production to inertial oscillation generation without explicitly knowing the wind. However, the wind sets the forcing stress profile, and this profile changes as the OSBL structure evolves (e.g., strong wind can deepen the ML). Thus, our analyses are only suitable for isolated wind events that do not strongly alter the structure of the OSBL. For stronger storms and longer integrations, temporally evolving stress profiles [Σ(z) → Σ(z, t)] can be used in the same way that traditional slab models have incorporated time-dependent MLD (D’Asaro 1985).

3. Results

We use both local and global observational datasets to assess the differences between the generalized slab model, the traditional slab model of Pollard and Millard (1970), and the PWP model (Price et al. 1986). We start by comparing these models with drifter observations of wind stress and surface currents in the Icelandic Basin. We then estimate the global impact of TLs on power input to near-inertial motions in the OSBL and TKE production.

a. Local analysis: Iceland Basin (NISKINe site)

The Office of Naval Research (ONR)-sponsored NISKINe in the North Atlantic is an ongoing Departmental Research Initiative (DRI) to investigate the generation, evolution, and dissipation of NIWs in the Iceland Basin. This region was chosen for the strong mesoscale eddy field and stormy weather.

As part of a NISKINe field campaign in the boreal summer of 2018, Klenz et al. (2022) discuss the deployment of a fleet of in situ Minimet surface drifters to observe the upper-ocean dynamics (i.e., surface to 15-m depth) in the Iceland Basin. The Minimet drifter (Centurioni 2018) is a robust platform for measuring surface meteorological data, and the Lagrangian nature of the measurements make it an excellent candidate for observing the power input to near-inertial motions, as discussed previously. Specific details pertaining to these observations can be found in Goni et al. (2017), Centurioni (2018), and Klenz et al. (2022).

On the 17 and 18 August 2018 in the Iceland Basin, a strong, impulse-like wind event occurred over a region of substantial mesoscale activity following deployment of the Minimet drifter fleet. Klenz et al. (2022) performed direct calculations of the power input to near-inertial motions associated with the wind event using the observations from the Minimets. We focus our local analysis on a 15-day period surrounding this wind event, from 10 to 25 August 2018.

Prior to the wind event, the step-like assumption of the slab model was violated since the MLD and TLD were 10 and 40 m, respectively (G. Voet 2023, personal communication), giving TLT/MLD = 3. We use these MLD and TLD values to parameterize a forcing stress profile for the generalized slab model. Closed form expressions for the MLTL and traditional stress profiles are provided in the appendix.

Klenz et al. (2022) calculate hourly surface wind stress from measurements of surface winds by Minimet 3 following Large and Pond (1981) [see Klenz et al. (2022, their Fig. 4) for drifter coordinates]. These wind stress data are used to force the traditional slab model (with a linear stress profile), generalized slab model (with a MLTL stress profile), and PWP (Fig. 4). Prior to the model runs, we apply a 24-h half-cosine Fourier filter to the wind stress time series to suppress the generation of low-frequency mean flows and isolate the near-inertial response. We follow the procedures in section 2 to calculate transports, inertial currents, wind work (total and available), TKE production, and the TKE fraction. Climatological stratification and ocean depths are obtained from the World Ocean Atlas (WOA) 2023 (Locarnini et al. 2023; Reagan et al. 2023) and version 24.1 of Smith and Sandwell (1997), respectively. Summations are truncated at 256 vertical modes. We use a constant Rayleigh damping coefficient of r−1 = 7 days in the analysis of each model, for consistency with Plueddemann and Farrar (2006). The Coriolis frequency f varies with drifter location in our model calculations, but the local vorticity is not considered. We configure and run PWP by following Plueddemann and Farrar (2006) and Alford (2020b). We assume negligible heat flux during the storm (Klenz et al. 2022).

The observed inertial velocities and those predicted by the MLTL model show excellent agreement in both magnitude and phase immediately following the wind event. PWP slightly underpredicts the observed inertial velocities, while the slab model overpredicts the inertial currents, which is expected for shallow MLDs (D’Asaro et al. 1995; Plueddemann and Farrar 2006). After a few inertial oscillations, a phase lag develops between the observations and model outputs and the oscillations detune. However, despite differences in phase, the inertial velocity magnitudes associated with the MLTL model output and observations remain consistent (Fig. 4).

The Iceland Basin is a region characterized by a highly energetic eddy field (Zhao et al. 2018; Thomas et al. 2020), so the phase lag may be caused by variability in local vorticity. The sharp nature of the detuning suggests the change is related to frontal behavior. In Fig. 3, we zoom in on the drifter track/location on 19 August 2018 (time of the phase change). Local vorticity is shown in the colored contours of the figure, obtained via satellite altimetry from the Data Unification and Altimeter Combination System (DUACS) and distributed by the Copernicus Marine Environment Monitoring Service (CMEMS) (Taburet et al. 2019). Because this method of vorticity calculation uses the assumption of geostrophic flow, it cannot resolve submesoscale vorticity and is thus limited to the mesoscale. However, the drifter track suggests that the Minimet encounters an oceanic front (∼60.27°N, 21.52°W) at the time when the phase offset appears in the inertial velocities. The drifter, initially traveling eastward, passes through a polarity change in relative vorticity that is coupled with a sea surface temperature (SST) gradient and shifts to westward propagation following its encounter with the front (note that SST data are obtained from the Minimet). The collection of these effects supports our hypothesis that the dephasing of the model and observations is related to a frontal change in local vorticity. Since the observations do not fully resolve submesoscale vorticity, we do not attempt to include these effects in the generalized slab model.

Fig. 3.
Fig. 3.

Minimet drifter track is displayed for 7–27 Aug 2018. The wind event under consideration occurred in the middle of this time series, on 17 August. Local vorticity is obtained via satellite altimetry [Data Unification and Altimeter Combination System (DUACS); Taburet et al. 2019] using the assumption of geostrophic flow and shown with the color contours. Along-track sea surface temperature (SST) is obtained from the Minimet and shown by the color of along-track data points. The drifter location on 19 Aug 2018 is shown with the cyan star. The green and red stars represent the drifter location on 7 and 27 Aug, respectively.

Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0167.1

The maximum values of instantaneous wind work calculated by the generalized slab model with the MLTL stress profile and PWP are consistent with the observations, while the slab model again overestimates the peak magnitude of Πavail (Fig. 4c). For each model, time-integrated wind work shows a variable degree of deviation from the observations (Fig. 4d). Values calculated using the MLTL stress profile show closest agreement with observations. Integrated wind work calculated via PWP and the traditional slab model exceeds observations in both cases. The increased available wind work in PWP is due to a bookkeeping difference. Note that PWP extracts TKE shear production through the damping stress, so all wind work is technically available to drive the inertial currents. In practice, the turbulence parameterization in PWP immediately dissipates a significant fraction of this available wind work.

Fig. 4.
Fig. 4.

The generalized slab model is employed to study the near-inertial response to an impulse-like wind event on 17–18 Aug 2018 at the NISKINe site in the Iceland Basin. Lagrangian Minimet surface drifters described by Klenz et al. (2022) provide in situ measurements of τ, usurf, and Πavail. (a) The observations of τ from Minimet 3 (Klenz et al. 2022) are shown and are used as the forcing input for the generalized slab model and PWP. Generalized slab model calculations are performed using both the slab and MLTL stress profiles. For consistency with Klenz et al. (2022, Fig. 7), MLD and TLD are set to 10 and 40 m, respectively. (b) Zonal inertial surface velocities. (c),(d) The instantaneous and time-integrated Πavail, respectively. (e),(f) The instantaneous and time-integrated PTL, respectively. In all cases, the purple line represents the slab model, the gold line represents PWP, the green line represents the MLTL model, and black line represents observations from Klenz et al. (2022).

Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0167.1

The wind event acts as an isolated, impulsive forcing event with minimal instantaneous wind work preceding or following 17–18 August (Fig. 4). In all cases (i.e., slab, PWP, MLTL, and observations), the peak instantaneous wind work and the peak wind stress magnitude occur on 17 August at 1600 UTC, suggesting that hourly resolution is too coarse to estimate the lag between the wind event and energy input to inertial motions.

There is a substantial burst of TKE production associated with interactions between shear stress and velocity shear in the TL surrounding the wind event that mirrors the burst in Πavail. Otherwise, PTL is generally small and nonzero. This intermittency in TKE production is consistent with observational evidence that momentum injection into the TL is also highly intermittent (Majumder et al. 2015). Although the TKE production is small compared to the wind work, the TKE production is always positive. After the 2-week example time series studied here, the final magnitude of time-integrated PTL is comparable to the final magnitude of time-integrated Πavail. Cumulatively over the course of this example 2-week time series, the total wind work is reduced by a factor of ∼2 as a result of TKE production in the TL.

The TKE dissipation rate (ϵ) can be measured directly using rapidly sampling microstructure shear probes (Gregg 1991; Thorpe 2005) and is thus more readily observable than TKE shear production (P). A precise transformation that allows ϵ to be determined from P and vice versa is a topic of ongoing research and beyond the scope of this paper (e.g., Zippel et al. 2022). However, we follow Alford (2020b) by using dimensional analysis and scaling to estimate the order of magnitude of ϵ from PTL that is calculated by the generalized slab model.

Ignoring advective and straining terms, TKE shear production is related to the dissipation rate (ϵ) and the buoyancy flux (Jb) via P = ϵ + Jb (Tennekes and Lumley 1972). Studying turbulent dissipation in the TL, Kaminski et al. (2021) show that a mixing efficiency Γ of 0.2 (defined Γ ≡ Jb/ϵ) can be used to relate buoyancy flux to dissipation. Finally, we note that the TKE production in our model has units of power input per unit area while ϵ has units of power input per unit mass. We may convert between the two using the density of seawater and the relevant length scale, which is in this case the TLT. Using values from the August 2018 wind event at the NISKINe site, we find ϵTL ∼ O(10−8), which is consistent with previous observations of ϵ in the TL, where ϵTL ∼ O(10−9–10−8) W kg−1 (Sun et al. 2013; Kaminski et al. 2021).

We calculate modal spectra for the slab and MLTL models and find that wind work in high modes attenuates much faster with increasing mode number for the MLTL model than for the slab model. Spectra are displayed in Fig. 5. Mode 3 is indicated in the figure to mark the classical division between radiative and dissipative modes (Alford 2020a). Following Fig. 4, we assert that available wind work calculated using the MLTL model provides wind work estimates more aligned with observations than the slab model. When viewed in tandem with Fig. 5, this suggests that the slab model overestimates the wind work in high modes, consistent with the findings of Plueddemann and Farrar (2006) and Alford (2020b). In the NISKINe test case specifically, we see that overestimation of Π becomes significant around mode 10.

Fig. 5.
Fig. 5.

Modal spectra for the slab and MLTL model are calculated for the NISKINe site (August 2018; 60°N, 22°W). The OSBL at the NISIKINe site has MLD of 10 and TLD of 40 m. (top) Spectral amplitude vs mode number is displayed. Amplitudes for the total wind work as calculated via slab model (Πtotslab), total wind work as calculated via MLTL model (ΠtotMLTL), and available wind work as calculated via MLTL model (ΠavailMLTL) are shown with the purple, black, and green lines, respectively. Note that total and available wind work are equal in the case of the slab model. All curves are normalized by the total amplitude of ΠtotalMLTL, for example, ∑n=1∞ϕnsϕn(0). (bottom) The cumulative modal amplitudes are displayed vs mode number for the total wind work calculated with the slab model (purple line), available wind work calculated with the MLTL model (green solid line), and TKE production in the TL calculated with the MLTL model (green dotted line). All curves are again normalized by the total amplitude of ΠavailMLTL. Normalization by ΠavailMLTL allows for easy viewing of the division of total wind work into IKE and TKE: in this example the sum of amplitudes for PTL and ΠavailMLTL are asymptotic to 0.22 with ∼30 modes and 0.78 with ∼20 modes, respectively (indicated on the right-hand y axis). The sum of amplitudes for Πtotslab is asymptotic to 3.12 with ∼180 modes, indicating that in the case of NISKINe, the slab model overestimates wind work by a factor of 3. Mode 3 is indicated in both panels to mark the classical division between radiative and dissipative modes (Alford 2020a).

Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0167.1

A secondary effect that results from the slab model’s overestimation of higher-mode wind work is that fewer modes are required to resolve Π when a TL is considered. For the NISKINe test case, total wind work via the MLTL model requires ∼40 modes to resolve, while total wind work via the slab model requires ∼150 modes to resolve (Fig. 5). This is easy to visualize from a mathematical analogy: in a modal decomposition of the slab model, we are reconstructing a step-like profile using smooth, continuous functions. This is analogous to reconstructing a square wave using a Fourier series, which will require infinite modes to reach closure. The TL serves to smooth the step-like discontinuity at the base of the ML, and allows us to resolve the dynamics with fewer modes.

Normalization of available wind work and TKE production by the total wind work highlights how the total power input by the atmosphere is partitioned. A larger fraction of total wind work goes to TKE production for higher modes (Fig. 5). Enhanced TKE production begins around mode 10, consistent with the slab model overestimation of Π. In our test case, available wind work is resolved with 25 modes and is 78% of the total wind work and TKE production is resolved with 40 modes and is 22% of the total wind work. Overall, total wind work calculated via the slab model exceeds total wind work via the MLTL model by a factor of 3.

In both models, the spectral peak lies in the dissipative regime at mode 4, with only 8% and 14% of energy contained in the radiative modes (1–3) for the slab and MLTL models, respectively (for discussion of dissipative versus radiative modes, see Alford 2020a). These dissipative spectra are reasonably expected considering the extremely shallow MLD of 10 m. However, the MLTL spectrum sees a red shift relative to the slab spectra. For deep MLs, the projection onto low modes is stronger than for shallow MLs (Alford 2020a). It follows that the stress profile’s impact on the modal projection is inversely proportional to the MLD (D’Asaro et al. 1995). Our test case at the NISKINe site, with such a shallow ML, provides a limiting case in this sense. Even for the 10 m-MLD here, there is less than a 2% difference between the modal amplitudes for the slab and MLTL stress profiles in the radiative modes (1–3). We conclude that a traditional slab model will perform well if only the low-mode dynamics are to be addressed, but one must proceed cautiously for higher modes. Beyond mode 3 deviations between the slab and MLTL models grow rapidly (Fig. 5), and the total number of modes used in analysis must be taken into careful consideration.

The cumulative modal amplitudes (Fig. 5, bottom) reveal how the MLTL stress profile results in a twofold reduction in the power input to inertial motions with comparison to the slab stress profile. First, because momentum is deposited beneath the ML, inertial currents are reduced, thereby reducing Πtot. This reduction of inertial currents is very pronounced in the Minimet dataset (Fig. 4b), with a corresponding ∼60% reduction in Πtot (Fig. 5, bottom). Second, the power imparted to inertial motions is given by Πavail, rather than Πtot. The available wind work incurs an additional ∼22% reduction as a result of TKE production in the TL (Fig. 5, bottom). For the NISKINe case study, the net effect is a ∼70% reduction in the power input to inertial motions.

b. Global Analysis: Geospatial characteristics of the TL and implications to TKE production

Data collected from Argo autonomous profiling floats throughout the lifespan of the program (1997–2023; Wong et al. 2020) are used to calculate the TL characteristics on a global scale. We restrict our analysis to high-resolution data (≤2 m in the vertical) because the base of the TL is difficult to resolve if the vertical resolution is too coarse (Johnston and Rudnick 2009; Helber et al. 2012; Sun et al. 2013). The local minimum method of Sun et al. (2013) is used to determine TLDs. The procedure picks the base of the TL as the depth corresponding to the first local minimum in stratification beneath the depth of maximum stratification. Sun et al. (2013) show this method to be reliable, but occasionally this method may fail if a local minimum does not appear until well below the seasonal thermocline. These cases can introduce anomalously large TLDs. For cases where the local minimum method fails, the depth of maximum stratification is used to characterize the TLD in an effort to keep estimates of TLD conservative. MLDs are taken from the climatologies developed by Holte et al. (2017). Together, observations of TLD and MLD allow for the calculation of TLT and the MLTL stress profiles at all grid points with high-resolution Argo data. We ignore the region within ±5° of the equator as slab-like models fail as f → 0 in the equatorial zone.

To assess the validity of a traditional slab model, we compute the relative TL thickness (TLTrel = TLT/MLD). We assert that if TLTrel ∼ O(1), the assumption of a negligible TL is violated. A global map of annual-mean TLTrel is shown in Fig. 6. The TLTrel mirrors many trends of MLD (see Holte et al. 2017), most notably that deep TLD and MLD appear in the mode water regions associated with the Southern Ocean, South Atlantic, Gulf Stream, and Kuroshio (Hanawa and Talley 2001). These patterns are consistent with the low-resolution estimates of absolute TLT thickness from Helber et al. (2012). In the bottom panel of Fig. 6, annual and seasonal median TLTrel values are plotted for each latitude. We use the median rather than the mean because the distribution of TLTrel is not Gaussian. Deviations from a normal distribution increase in regions where the ML is quite shallow. This creates outliers where TLTrel is extremely large, skewing the distribution of TLTrel.

Fig. 6.
Fig. 6.

Relative transition layer thickness (TLTrel, determined as TLTrel = TLT/MLD) is calculated using high-resolution Argo data and displayed for the global ocean. The pivot point of the diverging colormap is set at TLTrel = 1 so that grid points shaded red have TLT < MLD and grid points shaded blue have TLT > MLD. In the line plot below, annual and seasonal median values are given as a function of latitude. Error bars represent standard error, as the distribution of TLTrel is non-Gaussian. Depths less than 2000 m are omitted.

Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0167.1

Furuichi et al. (2008), Simmons and Alford (2012), Alford (2020a), and others have shown that there is strong seasonality to wind work that is tied to seasonal variability of the ML. This motivates us to assess seasonal variability of the TL. We follow Alford (2020a) in performing calculations for a global winter and summer. For a global winter we concatenate results from months 1–3 (January–March) in the Northern Hemisphere with months 7–9 (July–September) in the Southern Hemisphere. The opposite is done for a global summer. The idea behind the global seasonal extremes is to investigate bounding behavior of the TL. The magnitude of TLTrel is greater in the midlatitude regions during the wintertime, consistent with expectations of Helber et al. (2012).

We first use the global distribution of TLTrel calculated from individual Argo profiles (Fig. 6) to investigate the primary reduction in Πtot associated with the decrease in inertial current generation when momentum is deposited in the TL. In Fig. 7, we sum over 256 vertical modes to display the global distribution of annual-mean W=1−ΠtotMLTL/Πtotslab and a seasonal latitude-mean line plot. The reduction of Πtot is correlated to the geospatial distribution of TLTrel, with larger TLTrel producing greater disparities between Πtotslab and ΠtotMLTL. These disparities are magnified in some regions of the ocean, notably the Gulf Stream, north Indian Ocean, and Southern Ocean. We find that on the annual mean, using the MLTL stress profile rather than the traditional slab stress profile reduces total wind work by ∼15%–25%, with local reductions upward of 50% in extreme cases.

Fig. 7.
Fig. 7.

(top) The fractional, annual-mean increase in wind work related to increased surface currents in a slab model (W) is calculated from Argo data and shown globally to assess the impact the TL on wind work calculations. (bottom) Latitude-mean values of W as line plots, with error bars representing the standard error for each latitude bin. Depths less than 2000 m are omitted.

Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0167.1

The TKE fraction quantifies the secondary reduction in power input to inertial motions associated with the MLTL stress profile and is calculated via (18). The annual-mean P is shown in the map displayed in Fig. 8. The patterns in the TKE fraction map are similar to those in the TLTrel map (Fig. 6), where the TKE fraction is elevated in areas where TLTrel is large. As in Figs. 6 and 7, the Gulf Stream, north Indian Ocean, and Southern Ocean stand out with elevated P. Latitude-mean dissipation is calculated and displayed in the line plot of Fig. 8, with error bars representing the standard error of the mean. The latitude-mean plot of P in the bottom panel of Fig. 8 displays trends that do not identically mirror the trends in the latitude-mean TLTrel, contrary to expectation. Most notably, summertime TKE fraction exceeds that of the winter months throughout the ocean. On the global, annual mean, ∼30% of the total wind work is converted to TKE in the TL. In the high-latitude Northern Hemisphere, this percentage increases to ∼50%–70%, likely due to the thicker TLTrel observed in this region.

Fig. 8.
Fig. 8.

The TKE fraction calculated using the MLTL stress profile is shown for the annual mean. A value near 0 indicates that the there is little to no TKE production, consistent with the traditional slab model of Pollard and Millard (1970). A value near 1 indicates that nearly all of the wind work is diverted to turbulence production rather than exciting near-inertial motions. Latitude-mean values of P are shown in the line plot below the map, with error bars representing the standard error for each latitude bin. Depths less than 2000 m are omitted.

Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0167.1

4. Discussion

The curvature of the forcing stress profile has three important implications for the atmospheric power input to the internal wave band. First, the shape of the stress profile can spread momentum input over a greater depth range, reducing surface currents and total wind work (Πtot). Second, when the stress profile extends beneath the base of the mixed layer (ML), a fraction of the total wind work is converted into turbulent kinetic energy (TKE) production via the TKE fraction (PTL). Third, the shape of the stress profile sets the vertical wavenumber spectra, which influences the radiation of internal waves. The sum of these three effects manifests in a generalized slab model that offers improved agreement with observations over the traditional slab model (e.g., Fig. 4), and underscores the need to refine and improve models and measurements of τ(z) and ϵ(z) in the oceanic surface boundary layer (OSBL).

a. Calculation of total wind work

For the traditional slab model, wind work calculations often exceed observations due to the concentration of momentum in the ML. This issue has been recognized since the model’s introduction (Pollard and Millard 1970), and is exacerbated in locations with a shallow ML, as shown by D’Asaro (1995) and Alford (2020b). Including a TL allows near-inertial momentum imparted by the atmosphere to extend beneath the ML which decreases the calculated inertial currents. In cases of a shallow ML such as observed during NISKINe, this effect considerably improves agreement with observations. In the second panel of Fig. 4, the improved agreement for the case of a TL is self-evident, with inertial currents calculated using the stress profile of a traditional slab model exceeding observations by a factor of ∼2. We use Argo float data to estimate the magnitude of this effect on a global scale (i.e., Fig. 7) and find that wind work estimates calculated with the traditional slab model are biased high by a factor of 1.22 on the annual mean, globally. This is remarkably consistent with the overestimation bias of 1.23 reported by Alford (2020a).

b. TKE fraction

The TL is a region of strong shear, where the near-constant velocity profile of the ML adapts to the much smaller velocities characteristic of the stratified interior (Sun et al. 2013). When the forcing stress profile penetrates the TL, a nonzero fraction of the total wind work is converted into TKE production. This term is always zero in models that omit forcing in the TL and assume a linear stress profile in the ML. Using Argo float data to specify MLTL forcing stress profiles, we estimate the TKE fraction to be ∼0.3 on the annual mean, globally. In the high-latitude Northern Hemisphere, the TKE fraction increases to ∼0.5–0.7. When taken in tandem with the latest global, annual-mean estimates for near-inertial power input of 0.27 TW (Alford 2020b), these results suggest that 0.08 ± 0.01 TW of the total near-inertial power input are diverted to TKE production rather than generating near-inertial motions. This estimate could be greatly improved by collecting more observations of OSBL turbulence. For example, by equipping Argo floats with microstructure instruments (Roemmich et al. 2019).

The TKE fraction represents the relative fraction of the total wind work that is converted to TKE production when the TL is considered in the calculation. Despite the greater relative impact of the TL in the summer (Fig. 8), total wind work is much larger in the winter. Furuichi et al. (2008), Simmons and Alford (2012) find that wintertime wind work is 2–4 times greater in magnitude than in summertime. Thus, the total TKE production in the TL is substantially larger in the winter than in the summer, despite a smaller TKE fraction. For a simple numerical example, (Furuichi et al. 2008; Simmons and Alford 2012) estimate that wind work contributes an annual mean of ∼400 GW to the global ocean. Ignoring the spring and fall, for a naive argument, we take the middle ground and set winter wind work to 3 times that of summer, so that wind work in the summer contributes ∼200 GW while wind work in the winter contributes ∼600 GW. With these numbers and the seasonal TKE fractions calculated from Argo float data, it follows that ∼40 and ∼120 GW of the total wind work are converted to TKE production in the TL for the summer and winter, respectively.

c. Available wind work

Available wind work is the total wind work minus the energy lost to TKE shear production. Alford (2020b) estimated that global available wind work is only 39% of total wind work for PWP, while we estimate a value closer to ∼70%. The difference arises because Alford (2020b) computed power lost to turbulence by examining the change in potential energy due to ML deepening. ML deepening is driven by intense, short-lived, storms that dramatically alter the ML and TL. In contrast, the TKE shear production estimated here is based on ML and TL climatologies, such that our MLTL stress profile represents an “average” response to moderate winds. As stated earlier, our MLTL stress profile is not appropriate during storms that dramatically alter the ML and TL. For such storms, ML and TL statistics should be verified against observations to provide accurate wind-work predictions (e.g., Iceland Basin test case). However, in the high-latitude Northern Hemisphere where the TKE fraction is large, we find that the available wind work is reduced to ∼30%–50% of the total wind work, consistent with Alford (2020b).

A second difference in our estimation of available wind work is the interpretation of dissipation due to the Rayleigh damping term. Our ∼30% dissipation estimate of results from PTL, which arises from the forcing stress profiles alone with no consideration of how oscillations are damped after being set in motion. Additional analyses are necessary to better understand how the energy loss parameterized by Rayleigh drag is divided between internal wave radiation and additional TKE production. Estimates from Alford (2020a) are calculated using PWP estimates of changes in potential energy, which are driven by both “instantaneous” wind mixing and mixing induced after the wind forcing has stopped and the currents are spinning down.

d. Vertical wavenumber spectra

The analysis presented suggests that the slab model and MLTL produce comparable results for the low, radiative modes, but diverge considerably for the dissipative modes (4 to ∞). This indicates that the slab stress profile tends to produce more high-mode NIWs than the MLTL stress profile, suggesting that variability in the shape of the stress profile effects the vertical wavenumber spectra. This can have important implications to the ratio of energy which dissipates locally versus energy which radiates away from the generation site (Alford 2020a).

e. Relation to PWP

PWP and the generalized slab model both accurately predict inertial currents, which are necessary to calculate wind work (Fig. 4b). This suggests that the momentum removed from OSBL currents through ML deepening in the PWP model is of similar magnitude to the momentum injected beneath the mixed layer in the MLTL model, despite the different dynamics at play. A major strength of PWP is that, by accurately parameterizing turbulence dynamics, it can predict the vertical structure of currents using MLD and TLD from background stratification and wind stress alone. In contrast, the generalized slab model leverages observations to eliminate turbulence parameterizations (outside of the Rayleigh drag). In essence, the generalized slab model infers TKE shear production in the TL from observations of stress and currents (or MLD and TLD). From our results and conclusions here, future work may allow for improvements to the current implementation of the PWP algorithm by performing the same relaxation of assumptions and allowing for an arbitrary stress profile.

The excellent agreement between PWP and the generalized slab model currents in the Icelandic Basin implies that the turbulence parameterization in PWP is consistent with the observed MLD and TLD at that location. We reiterate that differences in available wind work (Fig. 4c) are primarily due to different bookkeeping conventions. PWP accounts for TKE shear production in the TL through a damping stress, while the generalized slab model accounts for TKE shear production through a forcing stress. Specifically, PWP estimates about 0.5 kJ m−2 of “excess” available wind work, which nearly matches the MLTL estimate of 0.6 kJ m−2 of PTL. These bookkeeping differences might be eliminated by applying wind stress in PWP using an MLTL stress profile rather than a linear (slab) profile. Such a modification would likely reduce regions of subcritical Richardson number, essentially “turning down” the PWP turbulence parameterization. However, we are uncertain how this modification would affect the accuracy of PWP current predictions, particularly during high winds.

We can use the PWP model as a control to estimate the viability of the generalized slab model’s inferred TKE production. Alford (2020a) calculated the traditional slab model’s overestimation bias factor of 1.23 by comparing the traditional slab results to PWP. We have calculated a comparable overestimation bias factor of 1.22 via comparison with the MLTL model. The two methods used here are completely different numerically, but produce nearly identical results. This implies that the MLTL and the PWP models both vertically spread momentum in the same way, despite the different approaches to the problem, and reinforces the importance of TKE production in the TL.

5. Conclusions

The generalized slab model developed here allows for the calculation of the ocean’s linear response to an arbitrary, nonlinear, depth-variable forcing stress profile. To introduce a first-order improvement upon the step-like, linear stress profile of the traditional slab model, we utilized a nonlinear MLTL stress profile inspired by D’Asaro et al. (1995) which injects momentum into the TL. Our analyses show that nonlinearities in the stress profile lead to a twofold reduction in power input to inertial motions. When the TL is considered, momentum is distributed to greater depths, reducing surface currents and inducing a primary reduction in wind work. A secondary reduction in wind work results from TKE production in the TL which reduces the amount of energy available to generate inertial currents in the OSBL. Direct comparison between Minimet observations in the Iceland Basin (Klenz et al. 2022), the traditional slab model, the generalized slab model with the MLTL stress profile, and the PWP model suggest that including the TL improves the slab model performance. Modeled TKE production in the TL in Icelandic Basin is consistent with other observations of turbulent dissipation in the TL (Sun et al. 2013). A global analysis of Argo autonomous profiling float data suggests that on the global, annual mean, ∼30% of the total power input to near-inertial motions is allocated to TKE production. When taken in tandem with the latest global, annual-mean estimates for near-inertial power input of 0.27 TW (Alford 2020b), these results suggest that 0.08 ± 0.01 TW of the total near-inertial power input are diverted to TKE production rather than generating near-inertial motions. We emphasize that our estimates of TKE production represent conditions of moderate winds and stress profiles based on climatology.

Acknowledgments.

I. A. Stokes was supported by ONR Grant N00014-22-1-2730. A. J. Lucas was supported by ONR Grants N00014-22-1-2730, N00014-22-1-2575, and N00014-21-C-1027. S. M. Kelly was supported by ONR Grant N00014-181-2800 and NSF Grant OCE-1635560. T. Klenz was supported by NSF Grant 1658302 and ONR Grant N000141812386. C. B. Whalen was supported by ONR Grant N00014-18-1-2598. A. F. Waterhouse was supported by ONR Grants N00014-18-1-2423 and N00014-22-1-2575. L. Centurioni and V. Hormann were supported by ONR Grant N000141812445. Minimet drifters used in this study were funded by ONR Grant N000141712517 and NOAA Grant NA150AR4320071 “The Global Drifter Program.” We thank the captain and crew of the R/V Neil Armstrong. Drifter velocities were calculated using J. M. Lilly’s MATLAB toolbox, jLab (Lilly 2021). The authors are grateful to Matthew Alford for his comments on a previous version of the manuscript, which led to an improved article. We thank Brewer et al. (2003) for providing the colormaps used in this work.

Data availability statement.

All shipboard and drifter data collected as part of the U.S. Office of Naval Research Near-Inertial Shear and Kinetic Energy in the North Atlantic experiment (NISKINe) are still in the process of being organized and archived. All data will be fully released to the public upon completion of this process. NISKINe data presented in this study are available from the corresponding author upon request. Argo data were collected and made freely available by the International Argo Program and the national programs that contribute to it (http://www.argo.ucsd.edu, http://argo.jcommops.org). The Argo Program is part of the Global Ocean Observing System. Argo float data and metadata are from Global Data Assembly Centre (Argo GDAC), SEANOE: https://doi.org/10.17882/42182. Satellite altimetry datasets are available from the CMEMS web portal (http://marine.copernicus.eu/services-portfolio/access-to-products/, last access: 29 July 2023) and the C3S data store (https://cds.climate.copernicus.eu, last access: 29 July 2023).

APPENDIX

Semianalytical Solution and Example Stress Profiles

a. Obtaining a semianalytical solution

Here we outline a procedure that can be used to obtain a semianalytical solution from the theory presented.

  1. Obtain the wind stress time series, stratification, and ocean depth for the study area.

  2. Solve the eigenvalue problem of (6) to obtain ϕn. This can be done using spectral methods consistent with Boyd (2001).

  3. Choose the structure of the stress profile Σ(z). We provide expressions for the traditional slab and the MLTL model in (A1) and (A2), but a form of Σ could be empirically derived from observations. One may also consider writing a new analytical structure function for Σ to represent an abnormal water column (e.g., double thermocline, inversions).

  4. Evaluate the integral in (10) to calculate ϕns. Unless the traditional slab model (A1) is used, this will need to be done numerically.

  5. Obtain ϕn(0). This is the value of ϕn at the ocean surface.

  6. Solve (8) numerically to obtain Un for each mode, then sum all modes to obtain total transport U. This can be done using a convolution method consistent with Gupta et al. (2019).

  7. Solve (14), (16), (17), and (18) to calculate Πtot, Πavail, PTL, and q, respectively.

b. Example stress profiles

1) Traditional, linear “slab” stress profile
Assume a linear ML stress gradient profile, consistent with the slab model of Pollard and Millard (1970). The conditions in a linear formulation are that the surface stress equals the wind stress, and is constant throughout the ML. Below the ML, the stress is zero. The vertical structure function for this stress profile can be expressed as
Σ(z)={1+z/Hmix for z>−Hmix,0 for −Hmix>z,
where MLD is denoted as Hmix. A sketch of this stress profile is shown with the black line in the right-hand panel of Fig. 1.
2) Quadratic MLTL stress profile
Assume a linear stress gradient profile in the ML, but include a quadratic taper from the bottom of the ML to the bottom of the TL to obtain the MLTL stress profile. The conditions for the MLTL stress profile are that the profile is continuously differentiable, the surface stress equals the wind stress, and both the stress and its vertical gradient go to zero at the base of the TL. Solving this system produces the structure function
Σ(z)={1+2(z/TLD)1+HmixTLD for z>−Hmix,1+2(z/TLD)+(z/TLD)21−(HmixTLD)2 for −Hmix>z>−TLD.
A sketch of this stress profile is shown with the blue line in the right-hand panel of Fig. 1.

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  • Alford, M. H., 2001: Internal swell generation: The spatial distribution of energy flux from the wind to mixed layer near-inertial motions. J. Phys. Oceanogr., 31, 2359–2368, https://doi.org/10.1175/1520-0485(2001)031<2359:ISGTSD>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Alford, M. H., 2020a: Global calculations of local and remote near-inertial-wave dissipation. J. Phys. Oceanogr., 50, 3157–3164, https://doi.org/10.1175/JPO-D-20-0106.1.

    • Search Google Scholar
    • Export Citation

  • Alford, M. H., 2020b: Revisiting near-inertial wind work: Slab models, relative stress, and mixed layer deepening. J. Phys. Oceanogr., 50, 3141–3156, https://doi.org/10.1175/JPO-D-20-0105.1.

    • Search Google Scholar
    • Export Citation

  • Alford, M. H., and M. Whitmont, 2007: Seasonal and spatial variability of near-inertial kinetic energy from historical moored velocity records. J. Phys. Oceanogr., 37, 2022–2037, https://doi.org/10.1175/JPO3106.1.

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  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    A schematic is shown for an example ML, TL, and the associated stratification and stress profiles. (left) Stratification data collected from a high-resolution Argo float in the Pacific Ocean on 1 Jan 2020 at 49.4°N, 129.3°W (World Meteorological Organization Argo float ID 4902432, cycle number 57). The mixed layer depth (MLD) and the transition layer depth (TLD) are calculated from the stratification data, using the density method of Holte and Talley (2009) and the local minimum method of Sun et al. (2013), respectively. The stratification data shown have been smoothed with a 15-point moving mean. (right) The stress profiles associated with the slab model and MLTL model are plotted vs depth in black and blue, respectively. Expressions for the vertical dependence of these profiles are derived in the appendix and given by (A1) and (A2), respectively.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    A flowchart is provided to visualize the pathways energy can take from wind in the atmosphere to mixing. The y axis is correlated to ocean depth and is superposed on a layered model of the atmosphere–ocean system (layers in descending order are atmosphere, ML, TL, interior). Boxes indicate processes in play, and box positions indicate which layer(s) the processes act on. The vertical positions of boxes within each layer are arbitrary. Processes highlighted in red are included for completeness but are not specifically addressed in the paper.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    Minimet drifter track is displayed for 7–27 Aug 2018. The wind event under consideration occurred in the middle of this time series, on 17 August. Local vorticity is obtained via satellite altimetry [Data Unification and Altimeter Combination System (DUACS); Taburet et al. 2019] using the assumption of geostrophic flow and shown with the color contours. Along-track sea surface temperature (SST) is obtained from the Minimet and shown by the color of along-track data points. The drifter location on 19 Aug 2018 is shown with the cyan star. The green and red stars represent the drifter location on 7 and 27 Aug, respectively.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    The generalized slab model is employed to study the near-inertial response to an impulse-like wind event on 17–18 Aug 2018 at the NISKINe site in the Iceland Basin. Lagrangian Minimet surface drifters described by Klenz et al. (2022) provide in situ measurements of τ, usurf, and Πavail. (a) The observations of τ from Minimet 3 (Klenz et al. 2022) are shown and are used as the forcing input for the generalized slab model and PWP. Generalized slab model calculations are performed using both the slab and MLTL stress profiles. For consistency with Klenz et al. (2022, Fig. 7), MLD and TLD are set to 10 and 40 m, respectively. (b) Zonal inertial surface velocities. (c),(d) The instantaneous and time-integrated Πavail, respectively. (e),(f) The instantaneous and time-integrated PTL, respectively. In all cases, the purple line represents the slab model, the gold line represents PWP, the green line represents the MLTL model, and black line represents observations from Klenz et al. (2022).

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    Modal spectra for the slab and MLTL model are calculated for the NISKINe site (August 2018; 60°N, 22°W). The OSBL at the NISIKINe site has MLD of 10 and TLD of 40 m. (top) Spectral amplitude vs mode number is displayed. Amplitudes for the total wind work as calculated via slab model (Πtotslab), total wind work as calculated via MLTL model (ΠtotMLTL), and available wind work as calculated via MLTL model (ΠavailMLTL) are shown with the purple, black, and green lines, respectively. Note that total and available wind work are equal in the case of the slab model. All curves are normalized by the total amplitude of ΠtotalMLTL, for example, ∑n=1∞ϕnsϕn(0). (bottom) The cumulative modal amplitudes are displayed vs mode number for the total wind work calculated with the slab model (purple line), available wind work calculated with the MLTL model (green solid line), and TKE production in the TL calculated with the MLTL model (green dotted line). All curves are again normalized by the total amplitude of ΠavailMLTL. Normalization by ΠavailMLTL allows for easy viewing of the division of total wind work into IKE and TKE: in this example the sum of amplitudes for PTL and ΠavailMLTL are asymptotic to 0.22 with ∼30 modes and 0.78 with ∼20 modes, respectively (indicated on the right-hand y axis). The sum of amplitudes for Πtotslab is asymptotic to 3.12 with ∼180 modes, indicating that in the case of NISKINe, the slab model overestimates wind work by a factor of 3. Mode 3 is indicated in both panels to mark the classical division between radiative and dissipative modes (Alford 2020a).

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    Relative transition layer thickness (TLTrel, determined as TLTrel = TLT/MLD) is calculated using high-resolution Argo data and displayed for the global ocean. The pivot point of the diverging colormap is set at TLTrel = 1 so that grid points shaded red have TLT < MLD and grid points shaded blue have TLT > MLD. In the line plot below, annual and seasonal median values are given as a function of latitude. Error bars represent standard error, as the distribution of TLTrel is non-Gaussian. Depths less than 2000 m are omitted.

  • <sc>Fig</sc>. 7.
    View in gallery
    Fig. 7.

    (top) The fractional, annual-mean increase in wind work related to increased surface currents in a slab model (W) is calculated from Argo data and shown globally to assess the impact the TL on wind work calculations. (bottom) Latitude-mean values of W as line plots, with error bars representing the standard error for each latitude bin. Depths less than 2000 m are omitted.

  • <sc>Fig</sc>. 8.
    View in gallery
    Fig. 8.

    The TKE fraction calculated using the MLTL stress profile is shown for the annual mean. A value near 0 indicates that the there is little to no TKE production, consistent with the traditional slab model of Pollard and Millard (1970). A value near 1 indicates that nearly all of the wind work is diverted to turbulence production rather than exciting near-inertial motions. Latitude-mean values of P are shown in the line plot below the map, with error bars representing the standard error for each latitude bin. Depths less than 2000 m are omitted.

Fig. 1.

A schematic is shown for an example ML, TL, and the associated stratification and stress profiles. (left) Stratification data collected from a high-resolution Argo float in the Pacific Ocean on 1 Jan 2020 at 49.4°N, 129.3°W (World Meteorological Organization Argo float ID 4902432, cycle number 57). The mixed layer depth (MLD) and the transition layer depth (TLD) are calculated from the stratification data, using the density method of Holte and Talley (2009) and the local minimum method of Sun et al. (2013), respectively. The stratification data shown have been smoothed with a 15-point moving mean. (right) The stress profiles associated with the slab model and MLTL model are plotted vs depth in black and blue, respectively. Expressions for the vertical dependence of these profiles are derived in the appendix and given by (A1) and (A2), respectively.
Fig. 2.

A flowchart is provided to visualize the pathways energy can take from wind in the atmosphere to mixing. The y axis is correlated to ocean depth and is superposed on a layered model of the atmosphere–ocean system (layers in descending order are atmosphere, ML, TL, interior). Boxes indicate processes in play, and box positions indicate which layer(s) the processes act on. The vertical positions of boxes within each layer are arbitrary. Processes highlighted in red are included for completeness but are not specifically addressed in the paper.
Fig. 3.

Minimet drifter track is displayed for 7–27 Aug 2018. The wind event under consideration occurred in the middle of this time series, on 17 August. Local vorticity is obtained via satellite altimetry [Data Unification and Altimeter Combination System (DUACS); Taburet et al. 2019] using the assumption of geostrophic flow and shown with the color contours. Along-track sea surface temperature (SST) is obtained from the Minimet and shown by the color of along-track data points. The drifter location on 19 Aug 2018 is shown with the cyan star. The green and red stars represent the drifter location on 7 and 27 Aug, respectively.
Fig. 4.

The generalized slab model is employed to study the near-inertial response to an impulse-like wind event on 17–18 Aug 2018 at the NISKINe site in the Iceland Basin. Lagrangian Minimet surface drifters described by Klenz et al. (2022) provide in situ measurements of τ, usurf, and Πavail. (a) The observations of τ from Minimet 3 (Klenz et al. 2022) are shown and are used as the forcing input for the generalized slab model and PWP. Generalized slab model calculations are performed using both the slab and MLTL stress profiles. For consistency with Klenz et al. (2022, Fig. 7), MLD and TLD are set to 10 and 40 m, respectively. (b) Zonal inertial surface velocities. (c),(d) The instantaneous and time-integrated Πavail, respectively. (e),(f) The instantaneous and time-integrated PTL, respectively. In all cases, the purple line represents the slab model, the gold line represents PWP, the green line represents the MLTL model, and black line represents observations from Klenz et al. (2022).
Fig. 5.

Modal spectra for the slab and MLTL model are calculated for the NISKINe site (August 2018; 60°N, 22°W). The OSBL at the NISIKINe site has MLD of 10 and TLD of 40 m. (top) Spectral amplitude vs mode number is displayed. Amplitudes for the total wind work as calculated via slab model (Πtotslab), total wind work as calculated via MLTL model (ΠtotMLTL), and available wind work as calculated via MLTL model (ΠavailMLTL) are shown with the purple, black, and green lines, respectively. Note that total and available wind work are equal in the case of the slab model. All curves are normalized by the total amplitude of ΠtotalMLTL, for example, ∑n=1∞ϕnsϕn(0). (bottom) The cumulative modal amplitudes are displayed vs mode number for the total wind work calculated with the slab model (purple line), available wind work calculated with the MLTL model (green solid line), and TKE production in the TL calculated with the MLTL model (green dotted line). All curves are again normalized by the total amplitude of ΠavailMLTL. Normalization by ΠavailMLTL allows for easy viewing of the division of total wind work into IKE and TKE: in this example the sum of amplitudes for PTL and ΠavailMLTL are asymptotic to 0.22 with ∼30 modes and 0.78 with ∼20 modes, respectively (indicated on the right-hand y axis). The sum of amplitudes for Πtotslab is asymptotic to 3.12 with ∼180 modes, indicating that in the case of NISKINe, the slab model overestimates wind work by a factor of 3. Mode 3 is indicated in both panels to mark the classical division between radiative and dissipative modes (Alford 2020a).
Fig. 6.

Relative transition layer thickness (TLTrel, determined as TLTrel = TLT/MLD) is calculated using high-resolution Argo data and displayed for the global ocean. The pivot point of the diverging colormap is set at TLTrel = 1 so that grid points shaded red have TLT < MLD and grid points shaded blue have TLT > MLD. In the line plot below, annual and seasonal median values are given as a function of latitude. Error bars represent standard error, as the distribution of TLTrel is non-Gaussian. Depths less than 2000 m are omitted.
Fig. 7.

(top) The fractional, annual-mean increase in wind work related to increased surface currents in a slab model (W) is calculated from Argo data and shown globally to assess the impact the TL on wind work calculations. (bottom) Latitude-mean values of W as line plots, with error bars representing the standard error for each latitude bin. Depths less than 2000 m are omitted.
Fig. 8.

The TKE fraction calculated using the MLTL stress profile is shown for the annual mean. A value near 0 indicates that the there is little to no TKE production, consistent with the traditional slab model of Pollard and Millard (1970). A value near 1 indicates that nearly all of the wind work is diverted to turbulence production rather than exciting near-inertial motions. Latitude-mean values of P are shown in the line plot below the map, with error bars representing the standard error for each latitude bin. Depths less than 2000 m are omitted.
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