OSOcean ScienceOSOcean Sci.1812-0792Copernicus PublicationsGöttingen, Germany10.5194/os-12-743-2016Dissipation of the energy imparted by mid-latitude storms in the Southern
OceanJouannoJulienjouanno@legos.obs-mip.frCapetXavierMadecGurvanRoulletGuillaumeKleinPatriceLEGOS, Université de Toulouse, IRD, CNRS, CNES, UPS, Toulouse,
FranceCNRS-IRD-Sorbonne Universités, UPMC, MNHN, LOCEAN Laboratory,
Paris, FranceNational Oceanographic Centre, Southampton, UKUniversity of Brest, CNRS, IRD, Ifremer, Laboratoire d'Océanographie
Physique et Spatiale (LOPS), IUEM, Brest, FranceJulien Jouanno (jouanno@legos.obs-mip.fr)1June201612374376913January201620January201614April201625April2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://os.copernicus.org/articles/12/743/2016/os-12-743-2016.htmlThe full text article is available as a PDF file from https://os.copernicus.org/articles/12/743/2016/os-12-743-2016.pdf
The aim of this study is to
clarify the role of the Southern Ocean storms on interior mixing and
meridional overturning circulation. A periodic and idealized numerical model
has been designed to represent the key physical processes of a zonal portion
of the Southern Ocean located between 70 and 40∘ S. It incorporates
physical ingredients deemed essential for Southern Ocean functioning: rough
topography, seasonally varying air–sea fluxes, and high-latitude storms with
analytical form. The forcing strategy ensures that the time mean wind stress
is the same between the different simulations, so the effect of the storms on
the mean wind stress and resulting impacts on the Southern Ocean dynamics are
not considered in this study. Level and distribution of mixing attributable
to high-frequency winds are quantified and compared to those generated by
eddy–topography interactions and dissipation of the balanced flow. Results
suggest that (1) the synoptic atmospheric variability alone can generate the
levels of mid-depth dissipation frequently observed in the Southern Ocean
(10-10–10-9 W kg-1) and (2) the storms strengthen the
overturning, primarily through enhanced mixing in the upper 300 m, whereas
deeper mixing has a minor effect. The sensitivity of the results to
horizontal resolution (20, 5, 2 and 1 km), vertical resolution and numerical
choices is evaluated. Challenging issues concerning how numerical models are
able to represent interior mixing forced by high-frequency winds are exposed
and discussed, particularly in the context of the overturning circulation.
Overall, submesoscale-permitting ocean modeling exhibits important delicacies
owing to a lack of convergence of key components of its energetics even when
reaching Δx= 1 km.
Introduction
Knowledge gaps pertaining to energy dissipation and mixing distribution in
the ocean greatly limit our ability to apprehend its dynamical and
biogeochemical functioning (globally or at smaller scale, e.g., regional) and
its role in the climate system evolution (Naveira-Garabato, 2012). For
example, the meridional overturning circulation in low-resolution global
coupled models is significantly altered by the parameterization for and
intensity of vertical mixing (Jayne, 2009; Melet et al., 2013).
A great deal of effort is currently deployed to address the issue but the
difficulties are immense: dissipation occurs intermittently, heterogeneously
and in relation to a myriad of processes, whose importance varies depending on
the region, depth range, season, proximity to bathymetric features, etc. In this
context, establishing an observational truth based on local estimates
involves probing the ocean at centimeter scale (vertically) with horizontal- and
temporal-resolution requirements that will need a long time to be met (e.g.,
MacKinnon et al., 2009 or DIMES program, Gille et al., 2012).
In order to make progress other (non-exclusive) approaches are being
followed. Well-constrained bulk-mixing requirements for certain water masses
can be exploited to infer mixing rates and, in some cases point to (or
discard) specific processes (de Lavergne et al., 2016). Alternatively,
in-depth investigations of dissipation and mixing associated with presumably
important processes are carried out (with the subsequent parameterization of
the effects in OGCMs (ocean general circulation models) being the ultimate
objective, Jayne, 2009; Jochum et al., 2013). This study belongs to the
latter thread. It is a numerical contribution to the investigation of
dissipation and mixing due to atmospheric synoptic variability (mid-latitude
storms) in the Southern Ocean.
Synoptic or high-frequency winds inject important amounts of energy into the
ocean that feed the near-inertial wave (NIW) field. A large part of the near-inertial energy (NIE) dissipates locally in the upper ocean, where it deepens
the mixed-layer and potentially has an impact on the air–sea exchanges and
global atmospheric circulation (Jochum et al., 2013). Nevertheless a
substantial fraction of the NIE also spreads horizontally and vertically away
from its source regions: beta dispersion propagates the energy toward lower
latitudes (Anderson and Gill, 1979), advection by the geostrophic circulation
redistributes NIE laterally (Zhai et al., 2005) and the mesoscale eddy field
favors the penetration of NIWs into the deep ocean by shortening their
horizontal scales (Danioux et al., 2008; Zhai et al., 2005), or through the
“inertial chimney” effect (Kunze, 1985).
Although the near-inertial part of the internal wave spectrum is thought to
contain most of the energy and vertical shear (Garrett, 2001), large
uncertainties remain on the amount of NIE available at depth for small-scale
mixing and whether/where it is significant compared to other sources of
mixing such as the breaking of internal waves generated by tides or the
interaction of the mesoscale flow with rough topography (e.g., Nikurashin et
al., 2013). The only present consensus is that NIE due to atmospheric forcing
does not penetrate efficiently enough into the ocean interior to provide the
mixing necessary to close the deep cells of the MOC (Meridional Overturning
Circulation)(Furuichi et al., 2008; Ledwell et al., 2011), below 2000 m.
On the other hand, the vertical flux of NIE at 800 m estimated by Alford et
al. (2012) at station Papa (in a part of the north Pacific not particularly
affected by storm activity) may have significant implications on mixing of
the interior water masses, depending on the (unknown) depth range where it
dissipates. Our regional focus is the Southern Ocean, where intense storm
activity forces NIW (Alford, 2003) that seem to have important consequences,
at least above 1500 m depth. Elevated turbulence in the upper 1000–1500 m
north of Kerguelen plateau has been related to wind-forced
downward-propagating near-inertial waves (Waterman et al., 2013); the clear
seasonal cycle of diapycnal mixing estimated from over 5000 ARGO profiles in
regions of the Southern Ocean where topography is smooth points to the role
of wind input in the near-inertial range (and NIW penetration into the ocean
interior; Wu et al., 2011).
The aim of this study is to (i) further clarify the mechanisms implicated in
NIW penetration into the ocean interior, (ii) more precisely quantify the
resulting NIE dissipation intensity including its vertical distribution and
(iii) better understand the current (and future) OGCM limitations in
representing NIE dissipation. (Findings on ii will be specific to the
Southern Ocean while we expect those on i and iii to be more generic.) For
that purpose, we perform semi-idealized Southern Ocean simulations for a wide
range of model parameters and different numerical schemes covering eddy
present to submesoscale-rich regimes.
Importantly, our highest resolution simulations adequately resolve the meso-
and submesoscale turbulent activity deemed essential in the leakage of NIE
out of the surface layers, as found in Danioux et al. (2011). In contrast to
this and other studies (Danioux et al., 2008), the realism of the ocean
forcing, mean state and circulation makes it more directly applicable to the
real ocean, provided that numerical robustness and convergence is reasonably
achieved.
The paper is organized as follows. The model setup is presented in Sect. 2
and the ocean dynamics and mean state that are simulated without storms are
described in Sect. 3. Section 4 describes the spatial and temporal
characteristics and consequences of the simplified NIW field generated by the
passage of a single storm (spin-down experiment). In Sect. 5
quasi-equilibrated simulations are analyzed in terms of pathways through
which the storm energy is deposited into the interior ocean and sensitivity
of the mixing distribution to storm parameters and numerical choices. In
Sect. 6, we characterize the long-term impact of the storms on the (large-scale) MOC, which turns out to be significant, mainly because of their effect
on and immediately below the ocean surface boundary layer. Section 7 provides
some discussion and Sect. 8 concludes.
Model
The numerical setup consists of a periodic channel configuration 2000 km
long (Lx, zonal direction) and 3000 km wide (Ly, meridional
direction) that aims to represent a zonal portion of the Southern Ocean
located between 70 and 40∘ S (Fig. 1). It is inspired by the
experiment described in Abernathey et al. (2011), which is mainly adiabatic
in the interior. We add three ingredients to our reference experiment deemed
essential to reach realistic levels of dissipation and whose consequence is
to enhance dissipation and mixing in the model ocean interior.
Three-dimensional representation of instantaneous temperature
(rectangular box, color scale ranges from 0 to 20 ∘C) and zonal
velocity (vertical section) for the reference simulation at 2 km after
30 years. The domain is a 2000 km long and 3000 km wide reentrant channel.
The configuration represents the Southern Ocean between 40 and
70∘ S. Average ocean depth is 3500 m with irregular bottom
topography, which limits the ACC (Antarctic Circumpolar Current) transport
and tends to enhance deep mixing. At the surface, synoptic storms are
included in the forcing. They generate NIWs, whose signature is visible in
the velocity section, as a layering of the mesoscale structures.
The bathymetry is random and rough. Horizontal scales of the reference
bathymetry range between 10 and 100 km and depths vary between 3000 and
4000 m. The bottom roughness, computed as the variance of the bottom height
(H), is 3×104 m2, which can be considered as intermediate
between rough and smooth and is representative of the roughness of a
large portion of the Southern Ocean topography (see map of roughness in Wu et
al., 2011). The inclusion of bottom topography aims to limit the
ACC
transport through bottom form stress (Rintoul et al., 2001) and to generate
deep and mid-depth mixing through vertical shear. Our horizontal resolution
≥1 km and the hydrostatic approximation used to derive the model
primitive equations do not permit the proper representation of upward
radiation and breaking of internal lee waves (Nikurashin et al., 2011).
Nevertheless, the deep flows impinging on bottom irregularities generate
fine-scale shear, which enhances dissipation and mixing close to the bottom,
as generally observed in the Southern Ocean (Waterman et al., 2013).
The surface and lateral forcing vary seasonally.
The objective is to reproduce a seasonally varying stratification and
mixed-layer depth. These seasonal variations are known to be important in the
formation process of mode waters and functioning of the overturning, since
surface cooling triggers mixed-layer convection.
The wind forcing includes idealized Southern Ocean storms.
These high-frequency winds induce intense near-inertial energy and mixing
into the ocean interior. From the analysis of scatterometer measurements,
Patoux et al. (2009) provided general statistics of the spatial and temporal
variability of the Southern Ocean mid-latitude cyclones for the period
1999–2006: most of the cyclones occurred between 50 and 70∘ S, have
a radius between 400 and 800 km and last between 12 h and 5 days.
Mesoscale cyclones lasting less than 4 days represent about 75 % of all
cyclone tracks (Yuan et al., 2009). The storm forcing design, detailed in
Appendix A and adapting the methodology followed by Vincent et al. (2012), is
based on these observations.
Configuration
The numerical code is the oceanic component of the Nucleus for European
Modelling of the Ocean program (NEMO; Madec 2014). It solves the primitive
equations discretized on a C-grid and fixed vertical levels (z coordinate).
Horizontal resolution of the reference simulation is 2 km. There are 50
levels in the vertical (with 10 levels in the upper 100 m and cells reaching
a height of 175 m at the bottom), with a partial step representation of the
topography. Sensitivity runs to both horizontal and vertical resolutions
(Δx between 1 and 20 km, Δx= 2 km with 320 vertical levels)
are an important part of this study. The model is run on β-plane with
f0=10-4 s-1 at the center of the domain and β=10-11 m-1 s-1. A third-order upstream biased scheme (UP3) is
used for both tracer and momentum advection, with no explicit diffusion. The
vertical diffusion coefficients are given by a generic length scale (GLS)
scheme with a k-ε turbulent closure (Reffray et al., 2015).
Bottom friction is linear with a bottom drag coefficient of
1.5×10-3 m s-1. We use a linear equation of state only
dependent on temperature with linear thermal expansion coefficient α=2.10-4 K-1. The temporal integration is achieved by a modified
Leap Frog Asselin Filter (Leclair and Madec, 2009), with a coefficient of 0.1
and a time step of 150 s for the 2 km experiments. Sensitivity to these
parameters and numerical choices are also performed.
Seasonal cycle of zonally averaged SST (a, f, ∘C),
mixed-layer depth (b, g, m) computed in both model and observations
with a fixed threshold criterion of 0.2 ∘C relative to the
temperature at 10 m, net air–sea heat flux (c, h, W m-2), and
the solar (d, i, W m-2) and non-solar (e, j,
W m-2) components of the air–sea heat flux. Climatological seasonal
cycles are built from observations (left column) and model outputs and
forcing. Observations include OAFlux products (Yu et al., 2007) for the
period 1984–2007 and de Boyer Montégut (2004) mixed-layer depth climatology.
Model data are from the last 10 years of the 2 km reference simulation
without storms.
Air–sea heat fluxes are built so as to represent the observed seasonal
evolution of the zonally averaged sea surface temperature and mixed-layer
depth in the Southern Ocean (Fig. 2a, b). The surface heat flux
Qnet is as follows:
Qnet=Qsolar+Qnonsolar, where Qsolar is the
shortwave heat flux and Qnonsolar the non-solar heat flux
accounting for the effect of longwave, latent, sensible heat fluxes and a
feedback term g (Tclim-Tmodel). This feedback term
depends on a sensitivity term g set to 30 W m-2 K-1 (Barnier
et al., 1995) and on the difference between Tclim, a SST climatology
that varies seasonally and Tmodel the model SST. The seasonal
amplitude of Qnet in the center of the domain is 200 W m-2
(Fig. 2h), a value close to the observations (Fig. 2c). Over the northern
150 km of the domain, the temperature is relaxed toward an exponential
temperature profile varying seasonally in the upper 150 m. The response of
the ocean to this forcing leads to a seasonal cycle of the surface
temperature (Fig. 2f), and a deepening of the mixed layer from 30 m in
summer to 150 m in winter (Fig. 2g), in good agreement with zonally averaged
observations of the Southern Ocean (Fig. 2a, b). It is worth mentioning
that the direct effect of a storm on the air–sea buoyancy flux (modulation of
the radiative, latent and sensible heat fluxes) is not explicitly accounted
for.
The background mean wind stress that forces the experiments without storms
is purely zonal:
τb=τ0sinπyLy,
where τ0=0.15 N m-2. In order to have exactly the same 10-year-mean wind stress between experiments with and without storms, the averaged
residual wind due to the storm passages is removed from τb in
the experiment with storms.
Two long reference experiments, one with storms and another without storms,
with a horizontal resolution of 2 km have been run for 40 years. For these
experiments, the model is started from a similar simulation without storms,
equilibrated with a 200-year long spin-up at 5 km horizontal resolution.
Unless otherwise stated, the last 10 years of the simulations are used for
diagnostics, excluding the northern 150 km band where restoring is applied.
Similar long-term simulations with a horizontal resolution of 20 and 5 km
have also been performed in order to determine meridional overturning
modifications with horizontal resolution (Sect. 7).
An experiment with a single storm traveling eastward through the center of
the basin over an equilibrated ocean has also been performed. Initial
conditions are taken from the 2 km horizontal resolution simulation (without
storm) at (day) 31 December of year 30 from the 2 km reference experiment
without storms. The storm is centered at the meridional position Ly/2 and
has a maximum wind stress of 1.5 N m-2. The ocean spin-down response is
analyzed for a period of 70 days (the storm is centered at days 5, starting
at day 3 and ending at day 7).
In order to assess the sensitivity of interior mixing to numerics and storms
characteristics, additional experiments have been run over shorter periods of
3 years, starting from year 30 of the 2 km reference experiment without
storms. These experiments are summarized in Table 1 and will be analyzed in
Sect. 5. The last 2 years of these experiments are used for diagnostics.
Although the model is not equilibrated after a period of 3 years, we have
verified in Sect. 5 that changes in terms of energy dissipation and mixing
diagnosed over this short period are significant.
Summary of numerical experiments.
NameΔxNb vert.DtHoriz. advStormsStorm speedTmaxlevels(Asselin coefficient)scheme(m s-1)(N m-2)Sensitivity to horizontal and vertical resolution 20 km nostorm20 km501200 s (0.1)UP3no20 km storms20 km”1200 s (0.1)”yes151.55 km nostorm5 km”300 s (0.1)”no5 km storms5 km”300 s (0.1)”yes””2 km nostorm2 km”150 s (0.1)”no2 km storms2 km”150 s (0.1)”yes””1 km nostorm1 km”60 s (0.1)”no1 km storms1 km”60 s (0.1)”yes””2 km nostorm_Z3202 km32050 s (0.1)”no2 km storms_Z320”32050 s (0.1)”yes””Sensitivity to horizontal advection scheme 2 km nostorm_QUICK”50150 s (0.1)QUICKno2 km storms_QUICK””150 s (0.1)QUICKyes””2 km nostorm_CEN2””100 s (0.1)CEN2no2 km storms_CEN2””100 s (0.1)CEN2yes””2 km nostorm_VFORM””100 s (0.1)VFORMno2 km storms_VFORM””100 s (0.1)VFORMyes””Sensitivity to storm characteristics 2 km storms_C0””150 s (0.1)UP3yes0”2 km storms_C5””””yes5”2 km storms_C10””””yes10”2 km storms_C15””””yes15”2 km storms_C20””””yes20”2 km storms_TAU-1””””yes1512 km storms_TAU-1.5””””yes151.52 km storms_TAU-3””””yes153One storm experiments 2 km onestorm_A””150 s (0.1)”yes151.52 km onestorm_B””30 s (0.1)”yes151.52 km onestorm_C””150 s (0.01)”yes151.5
The averaged total wind work in the 2 km experiment with storms is
16.8 mW m-2. This value is comparable to the 20 mW m-2 input
rates for the Southern Ocean estimated by Wunsch (1998). The contribution
from the near-inertial band is computed from instantaneous 2-hourly model
outputs, time filtered in the band {0.9,1.15}f following Alford et
al. (2012). Near-inertial wind work is 1.4 mW m-2 for the entire
domain and 2.2 mW m-2 in its central part
(1000 km < y < 2000 km). These values are in
agreement with Southern Ocean estimates from drifters (Elipot and Gilles,
2009; ∼2 mW m-2), ocean general circulation models (Rath et al.,
2014; ∼1 mW m-2) and slab mixed-layer models (Alford, 2003;
1–2 mW m-2).
Energy diagnostics
Energy diagnostics and precise evaluations of the energy dissipation in the
model are essential elements of our study. They are detailed below. The model
kinetic energy (KE) equation can be written as follows:
12ρ0∂tuh2︸KE=-ρ0uhuh⋅∇huh-ρ0uh⋅w∂zuh︸ADV-uh⋅∇hp︸PRES+ρ0uh⋅Dh︸εh+ρ0uh⋅∇zκv∇huh︸εv+Dtime,
where the subscript “h” denotes a horizontal vector, κv is
the vertical viscosity, Dh the contribution of lateral diffusion
processes and Dtime the dissipation of kinetic energy by the time
stepping scheme, which can be easily estimated in our simulations since it
only results from the application of the Asselin time filter. The dissipation
of kinetic energy by spatial diffusive processes is computed as the spatial
integral of the diffusive terms εv and
εh in Eq. (1):
Ev=∫∫∫ρ0uh⋅∇zκv∇zuh︸εvdxdydz=∫∫∫ρ0κv∂uh∂z⋅∂uh∂zdxdydz+∫∫uh⋅τs-uh⋅τbdxdy,Eh=∫∫∫ρ0uh⋅Dh︸εvdxdydz.
As mentioned before, we do not specify explicit horizontal diffusion since it
is implicitly treated by the UP3 advection scheme we use (see numerical
details in Madec, 2014). So the term Dh is evaluated at each time
step as the difference between horizontal advection momentum tendency
computed with UP3 and the advection tendency given by a non-diffusive
centered scheme alternative to UP3. Two options are the second-order and fourth-order schemes implemented in NEMO. The second-order scheme is non-diffusive but
dispersive. The fourth-order scheme in NEMO involves a fourth-order interpolation
for the evaluation of advective fluxes but their divergence is kept at second
order, making the scheme not strictly non-diffusive. Although the estimation
of UP3 horizontal diffusion depends on the scheme used as a reference, we
verify in Sect. 5 that the sensitivity of domain-averaged
εh to the choice of the second- or fourth-order scheme is much
smaller than that resulting from other parameter changes, e.g., small changes
in the characteristics of the atmospheric forcing.
Surface vorticity snapshot (s-1) over the entire model domain
at (day) 31 December of year 39 from the 2 km horizontal resolution experiment
without storms.
Ocean dynamics under low-frequency forcing
We first examine the dynamics and mean state of the experiment with
a horizontal resolution of 2 km and without storms in order to review the
background oceanic conditions within our zonal jet configuration. A snapshot
of the surface vorticity field (Fig. 3) illustrates the broad range of scale
resolved by the 2 km model and the ubiquitous presence of meso- and
submesoscale motions, including eddies and filaments. The slope of the annual
mean surface velocity spectrum in the meso- and submesoscale range is between
k-2 and k-3. The spectral slope varies seasonally (Fig. 4b), more
noticeably in the submesoscale range (60 km > λ; i.e.,
horizontal scales below 10 km), between k-3 during summer and k-2
during winter (for the meso- and submesoscale range in Fig. 4b, the thin dark
red line is superimposed on the thick dark red line). We interpret the
increase of submesoscale energy during winter as a direct consequence of
enhanced mixed-layer instabilities in response to a deep mixed layer
(Fox-Kemper, 2008; Sasaki et al., 2014).
Horizontal velocity variance in the 2 km reference experiments with
and without storms. (a) Kinetic energy power spectra as a function
of wavenumber (rad m-1) at 0, 1000 and 2500 m depth.
(b) Seasonal (summer is defined as December–January–February and
winter as June–July–August) kinetic energy power spectra at 0 and 1000 m
depth. Spectra are built using instantaneous velocity taken each 5 days of
the last 2 years of the 2 km simulations. Kinetic energy contained in the
wavelength ranges λ<60 km (c), 60 km <λ<600 km (d) and λ>600 km (e) as a function of
depth. In (b) and for wavenumber above
5×10-5 rad m-1, the winter surface spectra with and without
storms (dark red thin and thick lines) are superimposed, as well as the
summer and winter 1000 m spectra without storms (light and dark green thin
lines).
Model snapshots of a 2 km simulation at a mesoscale eddy location 2
days before (top) and 17 days after (bottom) the passage of a storm:
(a, f) zonal velocity (m s-1), (b, g) meridional
velocity (m s-1), (c, h) vertical velocity (m s-1),
(d, i) vertical shear (s-2) and (e, j) horizontal
strain (s-2). Snapshots after the passage of the storm (e–h)
are taken 50 km eastward in order to account for the advection of the core
of an anticyclonic mesoscale eddy. Isotherm are shown in the left panels
((a), (b), (f) and (g)) with contour
intervals of 1.25 ∘C from 2.5 to 10 ∘C. Before the passage
of the storm the simulation has been equilibrated without high-frequency
forcing, so the solution at day 2 is free of wind-forced NIWs. The snapshots
shown here correspond to day 2 and 22 in the time axis of Fig. 8. We choose
day 22 to leave enough time for the NIWs to reach the base of the
anticyclonic eddy.
Vertical profile of the 2 km experiments with and without storms
averaged over a period of 10 years and between Ly/3 and 2Ly/3 with
Ly the meridional length of the domain: (a) eddy kinetic energy
(m2 s-2), (b) rms of the vertical velocity (m s-1),
(c) temperature (∘C), (d) stratification (s-2)
and (e) vertical turbulent heat flux (W m-2). The eddy kinetic
energy is computed from anomalies to the zonal mean. Dashed lines are for the
experiment without storms. In (c) the horizontal lines indicate the
mean position of the mixed-layer base (computed with a fixed threshold
criterion of 0.2 ∘C relative to the temperature at 10 m as in
Fig. 2), and in (e) the vertical lines show the average net air–sea
heat flux (W m-2).
Eulerian mean streamfunction ψ(a, b), MOC
streamfunction diagnosed in isopycnal coordinates (c ,d) and
projected back to depth coordinates (e, f) from 10-year long 2 km
equilibrated simulations with (right) and without storms (left). Units are Sv
and the contour interval is 0.25 Sv. Temperature contours corresponding to
2, 4, 6, 8, 10, 12 and 14 ∘C are indicated in (c, d).
Positive cells are clockwise. The dashed lines in (c, d) represent
the 10, 50 and 90 % isolines of the cumulative probability density
function for surface temperature (following Abernathey et al., 2011), which
indicate how likely a particular water mass is to be found at the surface
exposed to diabatic transformation. Dotted lines in (e, f)
represent (from top to bottom) the 90, 50 and 10 % isolines of the
cumulative probability density function for mixed-layer depth. The vertical
dashed line at y=2850 km represents the limit of the northern boundary
damping area. Model transports have been multiplied by 10 in order to scale
them to the full Southern Ocean.
The energy contained at large scale and mesoscale (k<5×10-5 rad m-1) decreases with depth as indicated by the
spectra at 1000 and 2500 m (Fig. 4a). But note that the energy contained in
the wavenumber range 5×10-5<k<6×10-4 rad m-1
(i.e., the range associated with small mesoscale bordering with the
submesoscale) is larger at 2500 m compared to 1000 m. This is due to an
injection of energy at these scales by the rough topography. As shown by
instantaneous velocity sections in Fig. 5a and b, the horizontal scales of u
and v below 2500 m are much shorter than the typical scale of the upper-ocean mesoscale field. They correspond to the scale of the bathymetry, and
are responsible for increased horizontal shear in the deep ocean (Fig. 5e),
thereby contributing to the dissipation of the energy imparted by the winds
to the mean flow.
Vertical velocity rms is below 10 m day-1 over most of the water
column except near the bottom (i.e., below 2500 m) where it increases
substantially to ∼100 m day-1 (Figs. 5c and 6b). Although flat
bottom numerical solutions can also exhibit similar increases (Danioux et
al., 2008), the spatiotemporal scales of w near the bottom (e.g., see
Fig. 5c) suggest the importance of flow–topography interactions.
The average zonal transport in the reference experiment is ∼ 300 Sv.
Although the rough bathymetry strongly reduces the transport compared to
simulations with flat bottom (that reach ∼ 1000 Sv, not shown), the
absence of any topographic ridge and narrow passages does not allow us to
obtain the typical transport of ∼ 130–150 Sv observed in the
ACC (e.g., Cunningham et al., 2003). As discussed in
Abernathey et al. (2011), much of this elevated transport can be seen as a
translation of the system westward that is not expected to affect our
investigation of fine-scale dynamics and its effect on the transverse
overturning circulation. The average eddy kinetic energy (EKE) exceeds
0.05 m2 s-2 at the surface (Fig. 6a). Such a level of energy is
typical of ocean storm tracks of the Southern Ocean (e.g., Morrow et al.,
2010).
The clockwise cell of the Eulerian overturning streamfunction ψ
(Fig. 7a)
Throughout the paper, Eulerian and residual meridional
transports obtained from our 2000 km long channel are multiplied by 10 in
order to make them directly comparable to those for the full Southern Ocean,
whose circumference is ∼ 20 000 km.
illustrates the large-scale
response to the northward Ekman transport (that acts to overturn the
isopycnal) and the irregular return flow in the deep layers due to bottom
topography. This transport is largely compensated by an eddy-induced-opposing
transport, leading to a residual circulation (see e.g., Marshall and Radko,
2003). This residual MOC can be computed as the streamfunction
ψiso from the time- and zonal-mean transport in isopycnal
coordinates (e.g., Abernathey et al., 2011). In the lightest density classes
and northern part of the domain, the counterclockwise cell (negative, driven
by surface heat loss) is the signature of a poleward surface flow and
equatorward return interior flow, which can be interpreted in terms of mode
and intermediate water formation (see the bulge formed by the isothermal
layer between the 10 and 12 ∘C isotherms in Fig. 7e). The large
clockwise (positive) cell in the center of the domain consists of an
upwelling branch along the 1–4 ∘C isotherms and a return flow along
the 8–11 ∘C isotherms also contributes to mode water formation.
This clockwise cell exhibits a surface protrusion in the temperature range
8–14 ∘C (Fig. 7c) that resembles the upper-ocean MOC cell seen in
observations (Mazloff et al., 2013) but absent in the semi-idealized
experiments with annual mean surface forcings of Abernathey et al. (2011) and
Morrisonet al. (2011). In our experiments, the upper cell undergoes
major seasonal changes (not shown) again in agreement with observations by
Mazloff et al. (2013): clockwise near-surface transport is intensified in
boreal summer and fall, when the net heat flux is maximum and warms the upper
ocean, enhancing the transformation of the waters toward lighter density
classes. This upper cell is thus the result of the seasonal cycle of the
surface forcing. Our experiments do not account for the high latitude
anticlockwise cell associated with deep water formation because it is of no
concern for our purpose. In the 2 km reference case without storms, the
transport by the main clockwise cell of the MOC streamfunction results in a
realistic overturning rescaled value of 18 Sv (Table 2).
Maximum of the clockwise cell (as in the context of Fig. 7)
of the overturning streamfunction ψiso (Sv) averaged between
y=2000 km and y=2500 km. The streamfunctions have been computed using
10
years of 5-day average outputs from equilibrated experiments. Model
transports have been multiplied by 10 in order to scale them to the full
Southern Ocean.
As a first step, it is useful to consider a situation in which a single storm
disrupts the quasi-equilibrated flow described in the previous section so
that high-frequency forcing effects can be more easily identified. The storm
is chosen to travel eastward through the center of the domain. The experiment
is thoroughly described in Sect. 2 and the ocean spin-down response is
analyzed in Figs. 5, 8, 9 and 10 for a period of 70 days (the storm starts at
day 3 and ends at day 7).
Response of the ocean to the passage of a single storm:
(a) horizontal kinetic energy (log10 m2 s-2),
(b) horizontal kinetic energy in the NIW band (colors,
log10 m2 s-2) and difference of horizontal kinetic energy
between the simulation with storms and a reference simulation without storms
(iso-contours), (c) rms of the vertical velocity
(10-4 m s-1) defined as w2, where τθ=τmaxRr is the
horizontal average operator, (d)εv energy
dissipation due to vertical diffusion (W kg-1) and
(e)εh the energy dissipation due to horizontal
diffusion (W kg-1). These diagnostics are spatially averaged between
Ly/3 and 2Ly/3. The spatially averaged power spectra of the meridional
velocity (log10 m2 s-2 day-1) is shown
in (f) and has been computed using hourly data from day 0 to day 70.
The storm starts at day 3 and ends at day 7.
Temporal evolution of clockwise (CW) and counter-clockwise (CCW)
spectra as a function of vertical wavelength, computed from Wentzel–Kramers–Brillouin (WKB)-stretched
near-inertial velocities (a, b) for the single-storm experiment.
Units are m2 s-2 cpm-1. Near-inertial KE computed as a
function of time and depth from CW- and CCW-stretched velocities are shown
in (c) and (d). Units are m2 s-2.
NIW generation and propagation
After the passage of the storm, the horizontal currents between the surface
and 1500 m exhibit a layered structure with typical vertical scales of ∼100–200 m (Fig. 5f, g), which contrasts with the homogeneity of the
mesoscale currents before the passage of the storm (Fig. 5a, b). The layering
is similar to that observed in a section across a Gulf Stream warm core ring
by Joyce et al. (2013). It is associated with an increase of the horizontal
and vertical shear in the ocean interior (Fig. 5i, j). In agreement with
Danioux et al. (2011), we encounter that the storm intensifies the vertical
velocities in the whole water column (Fig. 5h). In response to the storm, KE
in the upper 100 m is strongly increased during 5 days (Fig. 8a). An
intensification of KE is also observed in the following days at depths below
500 m, indicative of downward propagation of the energy. A large part of
the additional energy injected by the storm occurs in the near-inertial range
(Fig. 8b): the space–time distribution of the near-inertial energy (colors)
matches rather well the difference of KE between the experiment with a storm
and a control experiment without a storm starting from exactly the same initial
conditions (contours).
The near-inertial energy propagates downward and its signature can still be
observed 60 days after the storm passage with two weak maxima: one at the
surface and another centered near 1500 m. Over the earlier part of the
simulation, we find downward energy propagation speeds
∼25 m day-1 in the upper 100 and ∼90 m day-1 between
100 and 1500 m. These values are higher than the 13 m day-1 average
propagation speed estimated by Alford et al. (2012) from observations at
station Papa, but are within the 10–100 m day-1 range estimated by
Cuypers et al. (2013) for NIW packets forced by tropical storms in the Indian
Ocean. Vertical velocities are generally intensified in the depth range where
stratification is weakest but the maximum of rms vertical velocities
qualitatively follows a similar behavior as near-inertial KE: it peaks at
2000 m depth a few days after the storm initiation, and then propagates
downward the following weeks (Fig. 8c).
Rotatory polarization of the near-inertial waves is useful to separate the
upward- and downward-propagating constituents of the waves. Rotatory spectra
(details of the methodology are given in Appendix B) of the stretched
profiles of velocity allow for a separation of the clockwise (CW) and
counter-clockwise (CCW) contributions to the energy as a function of time and
vertical wavenumber (Fig. 9a, b). Most of the energy is contained in the CW
part of the spectra, i.e., most of the energy propagates downward. While the
energy directed downward and contained in wavelengths between 1000 and
2000 m remains strong for about 30 days, the energy at short wavelengths
(< 500 m) is rapidly dissipated both for downward- and upward-propagating NIWs. The near-inertial KE computed from Wentzel–Kramers–Brillouin (WKB)-stretched CW and CCW
velocities (see Appendix B for details) are shown in Fig. 9c and d. Between days
20 and 30, the KE of CCW waves exhibits a maximum centered around
1500–2000 m. Because the highest topographic features only reach up to
3000 m depth; furthermore, since near-inertial velocities have been WKB
scaled,
we interpret this local maximum as the signature of interior reflection.
During the 5 days following the passage of the storm, we notice a slight
increase of both CW and CCW KE below 2500 m depth, suggesting NIW generation
at the bottom in response to storm forcing. Associated energy levels are
limited (< 10-2 m2 s-2) and no sign of vertical
propagation is observed so this process must be of minor importance, compared
to other flow–topographic interactions acting in the same depth range such as
lee-wave generation by the balanced circulation (Nikurashin and Ferrari,
2010).
Horizontal velocity frequency spectra computed at each depth and averaged
over the entire 70-day period of the experiment are shown in Fig. 8f. They
exhibit energy peaks at f, 2f and to a lesser extent 3f. The
near-inertial and super-inertial peaks are surface intensified but have a
signature throughout the water column. Waves with super-inertial frequency
arise after a few inertial oscillations and are exited by non-linear
wave–wave interactions (Danioux et al., 2008).
Dissipation of the NI energy
We now turn to the identification of the processes (either physical or
numerical) that dissipate the kinetic energy imparted by the storm. To this
end, the complete energetic balance of the single-storm experiment is
compared with that of a control experiment without a storm (Fig. 10). After
65 days, the experiment with storm returns to a horizontal kinetic energy
level identical to that of the control experiment (Fig. 10a). The e-folding timescale for the dissipation of vertically integrated KE imparted by the storm
is ∼ 20 days, but it only reaches 5 days for surface KE. The surface
value is consistent with estimates from drifter observations at similar
latitudes (Park et al., 2009). The different contributions of the right-hand side of
the kinetic energy equation (Eq. 1) that balance the input of energy by the
wind work are shown in Fig. 10b. First we note that the cumulated wind work
steadily increases after the storm passage (centered at day 5). This is due
to a slight strengthening of the large-scale eastward surface current in
response to the storm (not shown). This strengthening is a consequence of the
zonal current distribution as a function of latitude, which is not symmetric
with respect to y=1500 km, so the domain average additional zonal wind
work imparted by the storm is nonzero and positive. At day 70, 61.4 % of
the kinetic energy has been dissipated by diffusive processes in the upper
200 m, while 11.1 % has been dissipated between 200 and 2000 m and
4.3 % between 2000 m and the bottom (see Table 3). Bottom friction
(5.9 %) and pressure gradients (5.5 %) are also limited sinks for the
energy imparted by the storm. The cumulated contributions of horizontal
advection and Coriolis forces are small compared to the other terms
(< 1 %). The contribution of the Coriolis force to the energy
budget is not precisely zero due to the staggered location of u and v
points in our Arakawa C-grid. Most of the dissipation due to viscous
processes is achieved by vertical processes in the upper 200 m (80 %,
Fig. 10c). The maximum contribution of horizontal dissipation is between 200
and 2000 m where it is stronger than vertical dissipation (Fig. 10c).
Domain-averaged response of the ocean to the passage of a storm from
the same experiment already described in Figs. 5, 8 and 9. In order to
isolate the response of the storm, we show here the differences with a
reference experiment without storm and starting from exactly the same initial
conditions. (a) Horizontal kinetic energy (m2 s-2)
computed directly from model velocity (bold black) and indirectly from the
time integral of kinetic energy tendency computed online before (red) and
after (dotted red) Asselin time filtering. (b) Cumulated
contribution of the different terms of the KE equation (DIFF represent the
sum of both horizontal and vertical dissipations). (c) Cumulated
lateral (Eh) and vertical (Ev) energy dissipation
integrated in different depth ranges. (d) Cumulated dissipation of
energy by the Asselin time filter integrated in different depth ranges.
(e) Meridional distribution of cumulative wind work, viscous
dissipation, bottom friction, horizontal pressure gradients and Asselin
energy dissipation at day 70.
Cumulated energy dissipation at day 70 (see Fig. 10) relative
to a reference experiment without a storm, for three single-storm experiments
with different time step and Asselin time filter coefficient. Results for
the reference experiment described in Figs. 5, 8–10 are shown in the first
column.
Further insights on the distribution of viscous dissipation are obtained by
examining the temporal evolution of εv and
εh at all depths (Fig. 8d, e). It shows that the
largest kinetic energy dissipation rates are achieved by
εv in the upper 100 m during the 10 days following the
storm (Fig. 8d). Interestingly we note the presence of a maximum of
εv between 300 and 500 m depth between days 10 and 40,
with value of order 10-9 W kg-1. This is due to large
shear/dissipation values at depth in and below the core of anticyclonic
structures as illustrated in Fig. 5 and confirmed in Sect. 5. At these
intermediate depths, εh and εh are of
comparable magnitude. No significant near-bottom increase of
εv or εh is found during or after the
storm passage in Fig. 8d and e, although NIWs are generated at the bottom in
response to the passage of the storm (as seen in the previous section,
Fig. 9d). The levels of near-inertial energy below 2500 m depth remain 2 to
3 orders of magnitude lower than those found in the mixed-layer and are not
sufficient to significantly increase bottom dissipation.
The time filter contributes to dissipate 14 % of the energy imparted by
the storm, with dissipation well distributed in the entire water column
(Fig. 10d). This dissipation is highly dependent of the time step used in the
simulation (150 s) and “Asselin time filter” coefficient (0.1, the default
value used in most of the studies with NEMO). In a similar experiment with a
time step of 30 s, the contribution of the Asselin time filter falls to
3.4 % (see Table 3) and with an Asselin coefficient of 0.01 it falls to
1.5 %. This is coherent with temporal diffusion of the Asselin time
filter being proportional to the product of the Asselin coefficient by the
model time step (Soufflet et al., 2016). The temporal diffusion is divided by
5 when using a time step of 30 s instead of 150 s, and the temporal
diffusion is divided by 10 when using a coefficient equal to 0.01 instead of
0.1. In these two sensitivity experiments, the energy that is not dissipated
by the temporal filter is dissipated by lateral and vertical diffusion in the
entire water column, leading to a vertical distribution of total dissipation
(Asselin +εh+εv), which is similar
between experiments (see Table 3).
In terms of meridional distribution, most of the energy is dissipated below
the storm track (Fig. 10e). This questions the common hypothesis that a
significant part of the energy could be radiated away from the generation
area toward lower latitudes (e.g., Garrett, 2001; Zhai et al., 2004; Blaker et
al., 2012; Komori et al., 2008). In our configuration it appears that
vertical propagation and dissipation act much faster than horizontal
propagation.
Two-year mean KE balance (mW m-2) averaged over the
entire domain for the 2 km reference experiments with (left) and without
storm (right). The percentages give the fraction of total wind work that is
balanced by the terms of the KE equation. The second series of numbers and
percentages in the storm column refers to the storm - no storm differences.
As a reminder, where and through which mechanisms KE is dissipated, and in
particular the extra input of KE associated with storms, is the main focus of
our study. The dissipation of the energy imparted by the storms is now
investigated in the context of perpetual seasonally varying storm activity,
where time averaging can be used to reach statistical robustness. One storm
is formed every 10 days, travels at constant speed along a given latitude
(that changes for each new storm) and has a life cycle lasting 4 days and
composed of three phases (mature and linearly growing or decaying). The
seasonality of the storms is included by seasonally varying the maximum wind
stress of the storms from 0.75 N m-2 in austral summer to
1.5 N m-2 in austral winter (see details in Appendix A). We
successively focus on different related aspects of the simulations
energetics: the eddy-kinetic energy (EKE) distribution, the total KE balance, vertical distribution of KE dissipation and the
sensitivity of this dissipation to numerics.
EKE in the 2 km reference experiments
The additional input of energy by the storms modifies the levels of kinetic
energy in the flow. In the 2 km case without storms, the domain-averaged
10-year mean KE computed from zonally averaged velocities is
1.14×10-3 m2 s-2 and the EKE (i.e.,
1/2(u′2+v′2) where primed velocity anomalies are defined with respect
to zonally averaged velocities) is 5.21×10-3 m2 s-2.
When storms are included, both quantities increase (mean KE increases to
1.21×10-3 m2 s-2 and mean EKE increases to
5.34×10-3 m2 s-2). Besides this overall EKE increase,
EKE is decreased in the upper 300 m (Fig. 6a). Our interpretation is that
this arises owing to the storm reduction of the stratification (Fig. 6d). In
turn, this impacts the structure of the vertical modes and the inverse energy
cascade in a way that favors a less surface intensified distribution of EKE
with storms (Smith and Vallis, 2002). The small enhancement of EKE in the
range 1000–2000 m in the storm simulation is consistent with this
interpretation. There are other impacts of the storms: the rms of the
vertical velocity is increased by 1 order of magnitude in the whole water
column and reaches values on the order of 10-3 m s-1 (Fig. 6b);
the upper 100 m of the ocean get warmer and less stratified (Fig. 6c, d);
and the mixed-layer deepens by ∼30 m (horizontal lines in Fig. 6c).
Obviously, the heat budget is also affected with a +5 W m-2 increase
of the downward turbulent heat fluxes (Fig. 6e) and air–sea heat fluxes
(vertical lines in Fig. 6e).
The ability of near-inertial oscillations to propagate into the ocean
interior is affected by the mesoscale field (through the chimney effect, as
it will shown in Sect. 5.3) but is also intimately tied to the shrinking of
their horizontal scales so we expect to see non-trivial modifications of the
KE wavenumber spectra in the presence of storms. Near the surface the storms
impact is mainly perceptible at the lowest wavenumbers, the storms forcing
scale (Fig. 4a, e) or during summer at the submesoscale (Fig. 4b). This
larger influence of the storms during summer compared to winter in the
submesoscale range is explained by a larger impact of the storms on the
mixed-layer depth in summer compared to winter (not shown). During summer,
the mixed-layer is shallow (Fig. 2b, g) and sensitive to direct mixing by the
storms while during winter the mixed-layer is deeper and its depth is
controlled at first order by convective processes with storm passages having
a weaker influence. Modifications of the spectral slope (∼2.5) by the
storms are almost insignificant in the meso-/submesoscale range, where surface
dynamics energizes the flow, particularly at scales ∼10 km (wavelength
∼60 km) and below (Fig. 4a, d). The effect of storms at such fine
scales becomes pronounced below ∼300 m (Fig. 4c), where the surface
mode becomes attenuated
The typical vertical scale H(k) of the
surface mode at a wavenumber k is H(k)∼f/(Nk). Using
N=2×10-5 (see Fig. 6d) we find
H(k=10-4)= 225 m.
.
At 1000 m where the fine-scale energy associated with the NIW is largest
(Fig. 4c), the energy spectrum presents a bulge in the wavenumber range
10-4<k<10-3 rad m-1 that attests of the energy input at such
scales. This energy input is larger during winter than during summer
(Fig. 4b) in agreement with the storm forcing, which is more energetic during
winter. Fine-scales energization by the NIW can be seen down to ∼2500 m
(Fig. 4c) where it is confined to lower wavelength than at 1000 m (k>3×10-4 rad m-1). Limited signs of a large-scale
energy enhancement by the storms can be found at 1000 and 2500 m.
Kinetic energy dissipation (ε; W kg-1) as a
function of depth in experiments at 2 km with storms (continuous lines) and
without storms (dashed lines): total energy dissipation ε with
and without storms (a), dissipation due to vertical processes
εv and dissipation due to horizontal processes
εh(b), εh computed from a
second-order (UBS-C2) or fourth-order (UBS-C4) centered scheme (see text for
details) together with a 20-year mean and standard deviation of
ε for the 2 km reference experiment (c), and summer
(December–January–February) and winter (June–July–August) ε.
Profiles are computed using 5-day snapshots of the entire domain for a
2-year period. Position, strength and duration of the storms remain strictly
equal in the different experiments.
KE budget and dissipation in the 2 km reference experiments
Let us first examine in detail the KE balance (Table 4) in the two 2 km
reference experiments with and without storms. The KE balance in both
experiments are very similar, with overall wind work mainly balanced by the
work done by bottom friction (38.9 % without storms and 30.5 % with
storms), pressure work maintaining the system available potential energy
(32.2, 26.0 %) and vertical diffusion (23.4, 33.1 %). The KE balance
also indicates that the additional input of energy provided by the storms
(+3.64 mW m-2) is balanced at 90 % by dissipation
(-2.86 mW m-2 for horizontal and vertical dissipation to which one
should add the Asselin filter contribution) with pressure work and bottom
friction being secondary (-0.18 mW m-2 representing a
5 % contribution and -0.07 mW m-2 representing a 2 %
contribution). This is in stark contrast with the equilibration of the
low-frequency wind work feeding the balanced circulation.
Now let us focus on the spatial and seasonal distribution of the horizontal
and vertical KE dissipation terms εh and
εv. The vertical distribution of these terms are computed
using instantaneous outputs available every 5 days during the last 2-year of
the 2 km runs. This choice of a limited 2-year period is justified given the
smallness of the standard deviation of annual mean ε computed
using 20 years of simulation of the experiment with storms (Fig. 11c), e.g.,
compared to ε differences we present for different experiments.
As stated in Sect. 2, we estimate UP3 intrinsic horizontal diffusivity as the
difference between UP3 momentum tendency and the tendency given by a fourth-order advective scheme. The alternative use of a second-order advection scheme
produces very similar estimates of εh (Fig. 11c).
Overall energy dissipation (ε=εh+εv) in the reference experiments is
increased by 1 order of magnitude or more over most of the water column in
the presence of storms (Fig. 11a). Exception is found in the lowest 1000 m,
where dissipation is always strong because of the interaction of the
mesoscale and large-scale field with the topography. Without storms,
dissipation reaches a minimum of 3×10-12 W kg-1 between 1000
and 1500 m depth while the presence of storms increases the level of
dissipation to > 10-10 W kg-1 in this depth range, in
agreement with the results for the single-storm experiment (Fig. 8).
εh and εv (W kg-1)
distribution within composite cyclones (top) and anticyclones (bottom)
identified in the 2 km experiments without storms (left) and with storms
(right). The black iso-contours are isotherms from 2 to 8 ∘C and
σ/f iso-contours are shown in white (0.9, 0.95 and 0.98 σ/f),
with σ=f+ξ/2 the effective frequency and ζ the relative
vorticity. Composites are built using 10 years of 5-day-averaged model
outputs, between Ly/3 and 2Ly/3. A total of 8167 cyclone and 8878
anticyclone snapshots have been identified in the experiment without storms
and 7306 cyclone and 8037 anticyclone snapshots in the experiment with
storms.
The distribution of the dissipation between horizontal and vertical diffusive
processes and their respective sensitivity to the energy input by the storms
reveals some interesting behavior. First, vertical dissipation dominates in
the upper 200 m and (less clearly) below 3000 m, but in between, horizontal
processes account for most of the dissipation (Fig. 11b). This is
particularly true for the experiment with storms in which
εv is systematically less than 1/4 of
εh below 200 m. Second, there is an increase of
horizontal dissipation in the interior in response to the storms (Fig. 11b).
This is consistent with enhanced energy at short wavelengths (λ
< 60 km, Fig. 4a, c).
Since the air–sea heat fluxes and the strength of the storms follow a
seasonal cycle, we expect some seasonality of both near-surface and interior
dissipation. This is examined by comparing ε profile in summer
and winter (Fig. 11d). Values of ε in the upper 300 m display
large differences between summer and winter, in both experiments with or
without storms. Increased upper-ocean energy dissipation during winter is
explained by mixed-layer convection in response to surface heat loss. Below
300m, the experiment with storms is the only one that displays seasonal
variations of ε, with greatest values during winter. This is
consistent with observations by Wu et al. (2011), who observed a seasonal
cycle of diapycnal diffusivity (hence of ε) in the Southern Ocean
at depths down to 1800 m, although it reaches somewhat deeper (∼2500 m) in our solutions.
How do mesoscale eddies shape KE dissipation?
Mesoscale activity is known to affect NIW penetration into the ocean interior
(Danioux et al., 2011). In order to clarify the role of mesoscale structures
on energy dissipation distribution, an eddy detection method is used to
produce composite averages of dissipation, relative to eddy centers. The
identification of the eddies is based on a wavelet decomposition of the
surface vorticity field (e.g., Doglioli et al., 2007). Following Kurian et
al. (2011) a shape test with an error criterion of 60 % is used to
discard structures with shapes too different from circular. Since the Rossby
radius of deformation varies meridionally within the model domain, composites
are built with eddies located between Ly/3 and 2Ly/3, and with an area
larger than 400 km2. The barycenter is taken as the center of the eddies
and used as reference point to build the composites.
The general distribution of εh and εv
within composite eddies (Fig. 12) is in agreement with the vertical
distribution of domain-averaged ε discussed in the previous
section, with increased values of εh and εv near the surface and the bottom. But the composites also highlight
the impact of eddies on the distribution of εh and
εv. As discussed below the distribution of the kinetic
energy dissipation within eddies is very different depending on the presence
or absence of storms.
Kinetic energy dissipation (ε; W kg-1) as a
function of depth in experiments at 20, 5, 2 and 1 km horizontal resolution,
with storms (continuous lines) and without storms (dashed lines): total
energy dissipation ε with storms (a) and without
storms (b), dissipation due to vertical processes
εv with storms (c) and without
storms (d), dissipation due to horizontal processes
εh with storms (e) and without
storms (f), and the fraction of the total dissipation due to
vertical processes (εv/ε in %) (g
and h). As in Fig. 11, profiles are computed using 5-day snapshots
of the entire domain for a 2-year period. Position, strength and duration of
the storms remain strictly equal in the different experiments. The experiment
z320 has an horizontal resolution of 2 km but 320 vertical levels, ranging
from 1 m at the surface to 250 m at the bottom (below 2500 m depth the
vertical size of the cells is the same as in the 2 km reference experiment).
Kinetic energy dissipation (ε) and wind work as a
function of model resolution, in experiments with (continuous lines) and
without storms (dashed lines): (a) wind work and energy dissipation
integrated from surface to bottom (mW m-2), (d) energy
dissipation integrated from surface to bottom (decomposed into contributions
from ε, bottom friction and Asselin time filter; mW m-2)
and total dissipation ε (W kg-1) averaged in the depth
ranges 0–100 m (b), 100–400 m (c),
400–1000 m (d) and 1000–2000 m (e). Values are computed
using 5-day snapshots of the entire domain for a 2-year period as in Fig. 9.
Isolated dots represent ε for the 2 km experiment with 320
vertical levels. Wind work (a) and energy dissipation
contributions (d) have only been computed for the 20, 5 and 2 km
experiments.
Without storms, the distribution of either εh and
εv in the upper 1500 m shows that the border of the
cyclones and anticyclones are hot spots of dissipation, while the dissipation
at the center of the eddies is weaker than outside (Fig. 12a–d). This was
expected since horizontal strain and vertical shear are largest at the edges
of eddies and weak within the eddies. Near the bottom, dissipation is
increased below the cyclones centers (Fig. 12a, b) and decreased below the
anticyclones (Fig. 12c, d), owing to increased near-bottom velocities in
cyclones compared to anticyclones (not shown).
In the presence of storms (Fig. 12e–h), εv and εh peak at the base of the anticyclones with values higher than
10-9 W kg-1, in qualitative agreement with various observations
of NIW trapping at the base of the anticyclones (Joyce et al., 2013; Kunze et
al., 1995). The largest dissipation
is bounded by the contour σ=0.95f with σ=f+ξ/2 the
effective frequency. The compositing highlights the disproportionate
importance of anticyclones for NIW dissipation. The total area occupied by
the anticyclones that have been picked up by the eddy detection method
represents only 2.6 % each of the domain area, but concentrate the
interior KE dissipation at depth. Between 300 and 1500 m, 5 % of
εh and 17 % of εv is achieved
within identified anticyclones. Conversely, cyclone which statistically
occupy a similar area of the model domain are associated with only 4 % of
εh and 1.9 % of εv. The
statistical importance of anticyclones is further discussed in the
conclusion.
Sensitivity tests
How dissipation changes when key physical and numerical parameters are
varied is examined below.Horizontal resolution.
Energy dissipation is compared in experiments at
20, 5, 2 and 1 km horizontal resolution (Fig. 13). The sensitivity to
resolution strongly depends on the considered depth range. Near the surface
(0–100 m) the dissipation is almost not sensitive to the resolution
(Figs. 13a, b and 14b). This is coherent with the relatively weak variations
of the wind work from one resolution to another (Fig. 14a). But below
(100–400 m), experiments with or without storms show a decrease of
ε when increasing resolution (Figs. 13a, b and 14c). This
decrease is not related to modifications of the wind work (Fig. 14a) and
occurs in a depth range affected by upper-ocean convection. So it may mostly
result from the weakening of the dissipation due to upper-ocean convection
when resolution increases, as highlighted by the shallowing of the
mixed-layer depth (with storms and (without storms): 101 m
(93 m) at Δx= 20 km, 87 m (67 m) at Δx= 5 km, 80 m (59 m) at Δx= 2 km and 68 m
(53 m) at Δx=1 km). This would be in agreement with the
re-stratifying effect of the mesoscale and sub-mesoscale flow, which become
more efficient when resolution increases (e.g., Fox-Kemper, 2008; Marchesiello
et al., 2011).
In the depth range 400–3000 m, the sensitivity to resolution is highly
dependent on the presence or absence of storms. Without storms, a major
reduction of dissipation with increasing resolution is noticeable (Fig. 13b).
This reduction is of a factor 10 or more in the depth range 400–2000 m,
when going from 20 to 1 km resolution (Figs. 13b and 14c, e, f).
Concomitantly, the fraction of dissipation due to vertical shear increases
because that corresponding to lateral shear drops most rapidly (Fig. 13h). At
1km resolution, it is systematically above 20 % down to ∼2000 m and
reaches 50 % at 1500 m depth. This contrasts with the run at 20 km
where εv is never more than 7 % of the total
dissipation over the same depth range.
The behavior of interior dissipation with storms is strikingly different.
Dissipation changes with resolution are much more modest (in log scale). As
mentioned before, dissipation in the upper 100–400 m decreases when going
from Δx=20 to 1 km (Fig. 14c). Between ∼400 and 2000 m,
increasing resolution tends to increase dissipation (Figs. 13a and 14d, e).
At 20 km the mesoscale field is not well resolved and weaker; therefore, the
mesoscale near-inertial vertical pump is less efficient in transferring the
near-inertial energy into the interior; 5 km resolution changes total
dissipation significantly (e.g., from 4×10-11 to
1.2×10-10 W kg-1 in the depth range 1000–2000 m, Figs. 13a
and 14f). Changes are modest beyond Δx= 5 km. This is because
horizontal dissipation remains nearly unchanged and dominates total
dissipation. On the other hand, vertical dissipation exhibits interesting
changes in this resolution range. In particular, it keeps increasing and so
does its overall fraction in total dissipation. Also it develops a weak
relative maximum around 300–500 m at 1 and 2 km. We relate this maximum to
the one seen in dissipation composites for anticyclones (Fig. 12).
Near the bottom important changes also take place when increasing resolution:
vertical (horizontal) dissipation decreases (increases), which leads to a
slight decrease in dissipation by interior viscous processes. Instead,
dissipation by bottom friction increases significantly with resolution
(Fig. 14d). We are not sure how to interpret these bottom sensitivities,
especially since we do not properly resolve the processes implicated in
flow–topography interactions (Nikurashin and Legg, 2011).
Vertical resolution.
An experiment with 320 vertical levels has been
carried out in which vertical shears (and high-order vertical modes) are
better represented than with the reference 50 levels. The vertical thickness
of the cells increases from 2 m at the surface, 5 m at 500 m depth, 70 m
at 1000 m depth and 180 m near the bottom. The size of the cells below
2500 m are equal to the reference experiment so that the local
characteristics of flow–topography interactions are unchanged. The overall
dissipation ε is increased in the presence of storms in the interior
in the configuration with 320 vertical levels (Figs. 13a, b and 14c–e),
indicating that the downward propagation of the NIE is better resolved in the
high vertical resolution experiment with more NIE available at depth. A similar
increase of ε in the upper 100 m in the experiments with and
without storms (Fig. 14b) suggests that mixed-layer dynamics is profoundly
altered when changing the vertical resolution.
Advection schemes.
The reference experiment relies on an UP3
advection scheme (Webb et al., 1998). It is compared with three experiments
run with three widely used advection scheme: the QUICK (Quadratic Upstream
Interpolation for Convective Kinematics) scheme, which is the default scheme
of The regional oceanic modeling system (ROMS) model (Shchepetkin and
McWilliams, 2005) and also includes implicit diffusion; a second-order
centered scheme with a horizontal biharmonic viscosity of
-109 m4 s-2; and a second-order centered scheme with the
vector invariant form of the momentum equations (Madec, 2014) with the same
horizontal biharmonic viscosity. The implicit dissipation of UP3 and QUICK
take the form of a biharmonic operator with an eddy coefficient proportional
to the velocity (Ah=-|u|Δx3/12 with UP3 and
Ah=-|u|Δx3/16 with QUICK). Although QUICK is by
construction less dissipative compared to UP3, ε in both
experiments are very similar (Fig. 15a). With or without storms, the
second-order scheme in flux form (CEN2) or vector invariant form (VFORM)
leads to increased ε in the ocean interior with the increase
being the largest at the bottom (the energy dissipation profiles for the
second-order and the vector-form scheme are so close that they are
superimposed in Fig. 15a). Such distribution of the dissipation changes is
obviously related to the choice of a biharmonic coefficient of
-109 m4 s-2: characteristic velocities of 1.5 and
2 m s-1 are required for UP3 and QUICK schemes to match a biharmonic
diffusion coefficient of -109 m4 s-2. So near the surface
where currents are strong the explicit diffusion in the simulations with
second-order schemes is of same order as the implicit diffusion in QUICK/UP3
simulations, while at depth an explicit biharmonic operator with coefficient
-109 m4 s-2 overestimates the diffusion compared to
UP3/QUICK implicit diffusion. We also note a dissipation increase in the
depth range 1000–2000 m when using these schemes in the presence of storms.
Sensitivity closer to the surface is much more limited.
Maximum wind speed.
Stronger winds increase the energy dissipation in
the interior (Fig. 15c). Changes in dissipation levels take place from the
near surface down to 2500–3000 m, which again highlights that near-inertial
energy is able to propagate down to such depths. Dissipation changes induced
by modifications of the flow–topography interactions would also yield changes
in dissipation near the bottom, which is not the case, particularly when
comparing the 1 and 1.5 N m-2 experiments.
Storm speed.
The storm speed of the reference experiment was taken as
Cs=15 m s-1, a value close to the 12 m s-1 inferred
by Berbery and Vera (1996) in some parts of the Southern Ocean. But this
speed is expected to vary from storm to storm and impact the amount of energy
deposited into the near-inertial range as several studies have shown in
particular in the context of hurricanes (Price, 1981; Greatbatch, 1983,
1984). The response of the ocean to storms traveling at 20, 15, 10, 5 and
0 m s-1 is compared in Fig. 15b with other storm characteristics
(including trajectory) remaining unchanged. The storms travel exactly at the
same latitude and for the same duration as in the reference experiment with
Cs=15 m s-1. Above 3000 m depth, energy dissipation
increases with storm displacement speed until reaching the threshold of
15 m s-1 beyond which it reduces slightly. These results are
consistent with those of Greatbatch (1984) and in particular NIE is maximized
for a storm timescale L/Cs∼(2×500 km) /15 m s-1∼18 h close to the inertial timescale (2π/f), with
L the scale of the storm. Bottom dissipation is slightly enhanced (from
2×10-9 to 3×10-9 W kg-1) when storm speed
decreases, presumably as a result of more energy being injected in the
balanced circulation when storms move slowly.
Sensitivity of energy dissipation (ε) profiles to
numerics (a), storm speed (b) and storm
strength (c). Experiments with (without) storms are shown with
continuous (dashed) lines. The advective schemes tested in (a) are
UP3 (reference), QUICK, flux-form second-order centered advection scheme (CEN2)
and a vector form advection scheme (VFORM). The profiles of the latter two
(blue and green colors) are confounded in panel (a). Dissipation
induced by storms traveling at different speeds is tested in (c) for
propagation speeds of 0, 5, 10, 15 and 20 m s-1. In these experiments
the duration and the power of the storms are the same as in the reference
experiment (for which the storm propagation speed is 15 m s-1).
In (d), the sensitivity to the storm strength is tested by comparing
experiments with maximum wind-stress values equal to 1, 1.5 (reference) and
3 N m-2. All the sensitivity experiments are run at 2 km horizontal
resolution. They start from the same initial condition equilibrated without
storms, and they are run for 3 years. Profile are built using 5-day
snapshots of the entire domain for the last 2 years of the simulations.
Transformation rate (in Sv): total (a), contribution of
air–sea fluxes (b) and diffuse fluxes across isotherms for the 2 km
simulations without storms (c) and with storms (d). The
diffuse fluxes are separated into vertical (light gray) and lateral (black)
contributions. The dashed lines in (c) and (d) correspond
to transformation by diffuses fluxes below 300 m depth. Model transports
have been multiplied by 10 in order to scale them to the full Southern
Oceans.
More importantly we note that major relative changes in energy dissipation
levels occur in the ocean interior as U varies, with 1 order of magnitude
difference or more for storms traveling at 5 or 0 m s-1 compared to
storm traveling at 15 m s-1 in the depth range 400–2000 m. Important
changes are also found for U=10 m s-1, which further confirms the
subtlety of the ocean ringing and its consequences. In particular, note that
a 30 % increase or reduction of the storm displacement speed has more of
an
effect than a 30 % reduction in storm strength. It also suggests another
possible modus operandi for low-frequency variability in the atmosphere to
impact the functioning of the ocean interior through a modification of the
storm characteristics such as displacement speed.
Impact of the storms on the Southern Ocean MOC
KE dissipation and mixing are related in subtle ways. Given the profound
modifications of KE dissipation by high-frequency winds presented in the
previous sections we now assess the influence of the storms on the water-mass
transformations by examining the MOC sensitivity (Fig. 7). Storms increase
the clockwise cell intensity by 3 Sv that is a 16 % increase compared to
the experiment without storms. This shows that in our experiment the storms
contribute efficiently to the strength of the MOC. It is worth mentioning
that there are almost no changes in the mean Ekman drift as suggested by the
very similar Eulerian overturning streamfunction in the cases with and
without storms (Fig. 7a, b).
Both the MOC and the response of the MOC to the storms are sensitive to model
horizontal resolution (Table 2). Without storms, the maximum (and scaled)
value of the MOC decreases from 20.4 Sv at 20 km to 18.0 Sv at 2 km. This
is well related to the decrease of interior (below 100 m) kinetic energy
dissipation with resolution increase in the experiments without storms
(Fig. 13b). But when storms are included, the MOC increases with an amplitude
that depends on the resolution (+0.3 Sv at 20 km, +1.5 Sv at 5 km,
+3.0 Sv at 2 km), leading to transports that are relatively similar
between experiments (20.7 Sv at 20 km, 20.9 Sv at 5 km and 21.0 Sv at
2 km). Again this is in agreement with the sensitivity of the kinetic energy
dissipation to model resolution: the presence of storms increases the levels
of energy dissipation in the interior to a level, which remains broadly
constant at the different resolutions (Figs. 13a, 14).
The processes that dominate the changes of water-mass transformation in the
experiments with and without storms can be identified by means of an analysis
following Walin (1982), Badin and Williams (2013) and other. Water-mass
transformation rate G is defined as
G(ρ)=1Δρ∫Dair-seadA-∂Ddiff∂ρ
with Ddiff the diffusive density flux and
Dair-sea the surface density flux given by
Dair-sea=-αCpQnet,
where Qnet is the net surface heat flux, Cp the heat capacity of
the sea water, α the thermal expansion coefficient of sea water and
Δρ the density integration interval. The diapycnal volume flux is
directed from light to dense waters when G is positive. The computation of
the different terms is achieved following the technical details provided in
Marshall et al. (1999) with density bin Δρ of 0.1 kg m-3.
For easy comparison with previous results, the diagnostics are performed in
temperature space. As for momentum diffusion, the horizontal diffusion of
temperature is computed as the difference between UP3 temperature tendency
and the tendency given by a fourth-order centered scheme.
In the 2 km experiments without storms, the transformation by air–sea fluxes
is mainly from dense to light waters and peaks at -13 Sv near
6 ∘C (Fig. 16b; again the values here are scaled to the full
Southern Ocean). At this temperature, the transformation by diffusive
processes only reaches a modest -1 Sv (Fig. 16c) and the total
transformation rate (∼-14 Sv) is consistent with the 14.5 Sv of
meridionally averaged MOC transport centered at 6 ∘C (not shown).
The transformation by diffusive fluxes has two extrema near 4 ∘C and
12 ∘C, which correspond to temperatures where convection is more
active as suggested by the isolines of cumulative distribution of mixed-layer
depth in Fig. 7f or by the seasonal cycle of the mixed-layer depth in
Fig. 2b.
Overall, storms increase both the transformation by air–sea fluxes
(∼+3 Sv or +25 % at 6 ∘C) and diffusive fluxes
(∼+2 Sv or +130 % at 4 ∘C), leading to a ∼+3 Sv
total increase of water-mass transformation is the isotherm range
4–8 ∘C (Fig. 16a) that is consistent with the +3 Sv
strengthening of the main clockwise cell of the MOC. The change in the
air–sea fluxes is due to the feedback term that acts to restore model SST
toward its prescribed SST climatology. In the presence of storms, the
contributions from lateral and vertical diffusion are almost equal
(Fig. 16d), while without storms lateral diffusion dominates the water-mass
transformation (Fig. 16c). The fraction of transformation achieved below 300
m depth is very weak indicating that most of the diffusive transformation
process takes place in the near surface (Fig. 16c, d). On the other hand, an
important caveat is that only ∼20 % of the energy dissipated below
300 m is properly connected to mixing (through the k-epsilon submodel).
DiscussionModel realism and limitations
The realism of model dissipation is difficult to evaluate against
observations of dissipation rates because of spatial variability and temporal
intermittency in nature (see for example the longitude dependence of the
dissipation rate found by Wu et al., 2011, in the Southern Ocean; variability
at a finer scale is also important). With storms, mean interior dissipation
values at the highest resolution are in the range
1–10×10-10 W kg-1 depending on exact depth above 2000 m
and season. Such values are consistent with estimates from microstructure
measurements (Waterman et al., 2013; Sheen et al., 2013) or from release and
tracking of dye at mid-depth (Ledwell et al., 2011). However, they are on the
lower end of the ARGO estimates of Wu et al. (2011).
A source of uncertainty in comparing our simulations to observations is that
we lack internal-gravity wave generation by tides and we also misrepresent
the interaction between the geostrophic flow and bottom topography. Both of
these processes should significantly contribute to near-bottom dissipation
enhancement and their consequences around mid-depth may not be negligible.
Cabbeling and thermobaricity are other indirect sources of mixing that are
not taken into account in our study.
Assuming that Wu et al. (2011) estimates in regions with smooth bathymetry
primarily reflect dissipation of wind-input energy, we can nonetheless make
two important quantitative remarks. The vertical structure of storm energy
dissipation in our simulations is qualitatively consistent with their
observations: we find a factor 5–6 reduction in dissipation from 400 to
1800 m depth as they approximately do (their Fig. 3). Model seasonal
variations in ε also agree (note that we infer seasonal changes
of ε in Wu et al. (2011) from changes in diapycnal diffusivity,
assuming that subsurface stratification does not vary between seasons). Model
(respectively observations from Wu et al., 2011) winter to summer ε
ratios decrease from ∼2 (∼1.8) in the depth range
300–600 m to 1.6 (∼1.4) in the depth range 1300–1600 m.
These numbers agree within the error bars associated with observations by Wu et al. (2011). On the other hand, it is plausible that the slightly weaker
seasonal cycle systematically found in the observations arises from
dissipative contributions due to processes other than wind. The respective
roles of wind input and that of a distinct non-seasonally variable process on
dissipation could in principle be separated but model uncertainties and
limitations should also be kept in mind.
Near the bottom, our simulations generate dissipation at levels that are
essentially unaffected by synoptic wind activity (although this is less true
when storms travel slowly). ε reaches ∼5×10-9 W kg-1, a value which is not overly affected by numerical
resolution and turns out to be close to the values measured or inferred near-rough topography (Waterman et al., 2013; Sheen et al., 2013). This being
said, important reorganizations in the bottom 500 m from vertical to
horizontal dissipation as horizontal resolution increases suggest
cautiousness. So does the unrealistic representation of internal lee-wave
processes.
Energy pathways
Results by Nikurashin et al. (2013) suggest that the bulk of the large-scale
wind power input in the Southern Ocean is dissipated at the bottom by the
interaction of the mesoscale eddy field with rough (small-scale) topography.
Our simulations also show high energy dissipation at the bottom, but instead
of as in the rough experiment described in Nikurashin et al. (2013), for
which most of the energy imparted by the wind is balanced by interior viscous
dissipation, the wind input in our 2 km experiment without storms is balanced
by bottom friction (38.9 % associated with unresolved turbulence in the
bottom boundary layer), pressure work (32.2 %) and interior viscous
dissipation (23.4 %). This points out that we are not exactly in the same
regime as the one described in Nikurashin et al. (2013). This is probably
related to low roughness of our experiments compared to the rough experiment
in Nikurashin et al. (2013).
Using a global high-resolution model, Furuichi et al. (2008) estimate that
75–85 % of the global wind energy input to surface near-inertial motions
is dissipated in the upper 150 m. Similarly, Zhai et al. (2009) analyzing a
global 1/12∘ model found that nearly 70 % of the wind-induced
near-inertial energy at the sea surface is lost to turbulent mixing within
the top 200 m. Our results are in qualitative agreement with these studies:
in our high-resolution simulations only ∼65–70 % of the overall
energy imparted by the storm is dissipated in the upper 200 m (65 % in
the one storm experiment; see Table 4; 70 % in the multiple storm
experiment, not shown). Note though that, in contrasts to Furuichi et
al. (2008), who base their estimate on the near-inertial response of the wind
energy input, we do not separate the balanced and unbalanced response to the
storms. A substantial part of the additional wind work imparted by the storms
is not near-inertial, as revealed by the 1.4 mW m-2 near-inertial wind
work in the experiment with storms, which is only a fraction of the
+3.6 mW m-2 total wind work increase compared to the experiment
without storms. Since the balanced response to the storms does not follow the
same pathway toward dissipation (see below), such differences between our
results and Furuichi et al. (2008) are not unexpected.
Using a 1/10∘ model of the Southern Ocean, Rath et al. (2013) found
that accounting for the ocean-surface velocity dependence of the wind stress
decreases the near-inertial wind power input by about 20 % but also damps
the mixed-layer (ML) near-inertial motions leading to an overall
∼40 % decrease of the ML near-inertial energy. Overall, this damping
effect is found to be proportional to the inverse of the ocean-surface
mixed-layer depth. In our set of simulations, we do not include any
wind-stress dependence on ocean-surface velocity, which remains a debated
subject (Renault et al., 2016). Our main motivation for doing so was to
ensure that the mean wind stress remains the same between the different model
experiments that have been performed in this study. Nevertheless, we should
keep in mind that we miss a potentially important dissipative process for the
NIWs. The vertical turbulence model we use does not include an explicit wave
description so the surface wave mixing effect is parameterized and non-local
wave breaking, Stokes drift or Langmuir cells are not considered. These
processes modulate the momentum and energy deposited into the ocean as well
as near-surface dissipation rates. For example, the analysis of a coupled
atmosphere–wave–ocean model simulating hurricane conditions suggests that
the Stokes drift below the storm can contribute up to 20 % to the
Lagrangian flow magnitude and change its orientation (Curcic et al., 2016).
These processes certainly impact the near-inertial wind energy input and
distribution of its dissipation, and would deserve further attention, perhaps
using a more realistic (regional) setup.
Finally, our experiments provide an interesting perspective on the
dissipation of the energy associated with the slow versus NIW part of the
flow. The ways the energy imparted to the ocean by high and low frequency
winds are balanced differ markedly as one may have expected. Wind work
imparted by the storms is mainly balanced by viscous dissipation
(> 80 %), mainly in the upper ocean and to a lesser extent in
the interior. Bottom friction (∼5 %) and pressure work (∼5 %) play a minor role while these two terms are key in the
equilibration of the low-frequency part of the circulation (note that the
loss term associated with pressure gradient forces represents the potential
energy source due to Ekman pumping). Perhaps more surprisingly, total
interior dissipation in the simulation with and without storms present
distinct sensitivities with respect to resolution. As horizontal/vertical
resolution increases storm energy dissipation tends to diminish within a few
hundred meters below the mixed layer base but increases farther down.
Conversely, dissipation of the balanced circulation sharply decreases with
increasing resolution over a broad range of depth in the ocean interior, from
below the mixed layer down to 3000 m depth. It is also the situation where
convergence is least clear in the range of resolutions that we explore. Even
Δx=1 km resolution may still be insufficient to adequately resolve
fine-scale dissipative processes affecting the balanced flow (Vanneste,
2013). In any event and far from topographic features, dissipation of the
balanced flow, which is robustly 1 to 2 orders of magnitude smaller than
dissipation of the NIE below 300 m depth, is unlikely to have a substantial
effect on diapycnal mixing in the Southern Ocean interior.
Conclusions
Kinetic energy (KE) dissipation and its effect on ocean mixing are a subject
of intense research. The aim of this study is to investigate the fate and the
overall impact of the energy imparted by the storms in the Southern Ocean.
The set of semi-idealized numerical simulations we use to this end allow us
to explore and to identify the limitations faced by the general/regional
ocean modeling community in the numerical representation of these processes.
We also provide an additional perspective on the MOC sensitivities (to high-frequency winds) in a semi-idealized representation of the Southern Ocean
that shares important characteristics with the ones used in Abernathey et
al. (2011), Morrison and Hogg (2013; MOC sensitivity to the mean wind stress)
or Morrison et al. (2011; sensitivity to surface buoyancy forcing).
The main oceanic response to storm forcing involves the generation and
downward propagation of NIWs. While ∼ 60 % of the energy imparted by
the storms is dissipated in the upper 200 m, a substantial part propagates
and dissipates at greater depth (∼20–30 %). The NIWs that
penetrate downward have short horizontal wavelengths (λ < 60 km), high vertical shear and horizontal strain variance,
contributing to their dissipation before they reach the bottom.
In our simplified simulations near-inertial oscillations are the dominant
source of mixing down to 2000–2500 m depth. Our model results also confirm
the conclusions of several previous numerical and observational studies:
atmospheric synoptic variability and its associated internal energy wave
activity generation is required to explain the levels of mixing observed in
the interior ocean away from rough bathymetric features. This additional
input of energy becomes critically important as the resolution increases and
viscous dissipation of the balanced circulation vanishes (without storms a
2 orders of magnitude reduction of interior dissipation is found when going
from Δx= 20 km to Δx= 1 km). The inclusion of storms
leads
to comparatively minor sensitivities of interior dissipation to model
resolution. This has profound consequences on the MOC sensitivity to model
horizontal resolution: while without storms the strength of the clockwise
cell of the MOC decreases when resolution increase (also observed in Morrison
and Hogg, 2013), the introduction of storms tends to level off the
differences between resolution and to produce a slight increase of the MOC
with increasing resolution (Table 2).
We have shown that anticyclones play a disproportionate role as a conduit to
the interior ocean dissipation. This could certainly be anticipated from the
several studies describing the presence and dissipative fate of NIW packets
in anticyclonic structures. We are able to characterize this statistically.
We found that between 300 and 1500 m, 17 % of the dissipation achieved
by vertical processes occurs within identified anticyclones (versus 2 %
within identified cyclones). This estimate is a conservative figure because
we use a stringent eddy identification procedure.
Even with the storms included, dissipation below 200–300 m is too modest to
substantially influence water-mass transformation (Sect. 5). This result
should however be considered cautiously. Increased resolution (particularly
horizontal) beyond the range we explored may lead to further enhancements of
dissipation in the depth range 200–500 m. More importantly perhaps,
horizontal dissipation (which results from implicit numerical diffusion in
the advection of momentum) is dominant below the mixed layer and its effects
on diapycnal mixing may not be adequately represented. Indeed, it does not
contribute to the calculation of vertical mixing of temperature and its
connection with horizontal mixing (also resulting from implicit numerical
diffusion) is unknown
Note that in the case where diffusion and
viscosity operators and coefficients are explicitly prescribed no consistency
between KE dissipation and horizontal mixing of temperature is enforced
either.
. The relation between energy dissipation and mixing is a subject of
intense research. Ground-truth exists from direct numerical simulations (DNS)
or lab experiments (Shih et al., 2005; Ivey et al., 2008) but their
utilization is not straightforward here because of the large-scale gap with
our simulations in terms of resolved length scales (our ∼1 km
horizontal resolution places us several orders of magnitude away from the
isotropic regime).
The effect of storms is obviously most significant in the upper ocean. A
Walin analysis highlights this role and the consequences on large-scale ocean
dynamics. In our simulations storms significantly modify the vertical
buoyancy flux, air–sea heat fluxes (which are interactive) and MOC intensity
(+16 %). Although the settings have differences, an instructive
comparison consists in estimating the change in mean wind stress required to
increase the upper-MOC cell (the only one we simulate) by 16 % in the
sensitivity experiments carried out by Abernathey et al. (2011). Their Fig. 5
indicates a change from 0.20 to ∼0.23 N m-2 (+15 %) is
needed when interactive air–sea fluxes are used. This further confirms the
importance of synoptic winds. The effect of storms expressed in terms of
change in net air–sea heat fluxes is less dramatic (+5 W m-2) and
well within uncertainties (Wainer et al., 2003). On the other hand, the
fluctuations of heat fluxes due to storms have not been considered in our
study and their impact should be further investigated.
Important conclusions of this study also concern the numerical and physical
sensitivities of the NIE fate. Our analyses and sensitivity runs highlight
the effect of the Asselin filtering, of the numerical scheme employed for
advection, of numerical resolution, horizontal and to a lesser extent
vertical. Although ε changes with horizontal resolution tend to
level off when approaching Δx= 1 km, a more subtle lack of
convergence is patent. Most importantly, the respective contributions of
horizontal and vertical dissipations to ε still exhibit major
changes between Δx= 2 km and to Δx= 1 km, mainly in the
depth range 200–500 m. The reason why this may be of concern is that
vertical and lateral dissipation have a priori very different consequences in
the model, in ways that are difficult to reconcile with the isotropy of
microscale turbulence measured in the real ocean. In the model, vertical
dissipation is an essential component of the vertical turbulent closure and
modulates diapycnal mixing. Although lateral dissipation may also be
accompanied by diapycnal mixing (near fronts), existing ocean models have not
been widely evaluated or tuned in this regard. Ongoing efforts are aimed at
reducing lateral diapycnal diffusion in OGCMs but it is unclear down to which
level this should be pursued. The tendency found over the range of Δx
explored in this study suggests a robust εv increase to
the detriment of εh at depths between 200 and 500 m. The
strength of the diapycnal mixing that takes place in this ocean range is
important as demonstrated by the MOC sensitivity analysis in Sect. 6. Further
efforts to approach convergence and diminish grid anisotropy for problems
resembling the one studied here would be needed.
The modifications of the Southern Ocean atmospheric circulation have
motivated many studies on the response of the Antarctic Circumpolar Current
and Southern Ocean overturning to increases in mean wind stress (e.g.,
Abernathey et al., 2011; Hogg et al., 2015), the general conclusion being
that the eddies strongly limit the sensitivity of the ACC transport and
Southern Ocean MOC to wind increase (e.g., see the review by Gent, 2016). But
besides zonal wind strengthening, changes are also observed in the storm
track activity (see the review by Ulbrich et al., 2009). The evolution during
the last 50 years consists of a concomitant decrease of the overall number of
Southern Ocean cyclones and increase of their strength. This tendency is
expected to continue under warming climate. Alford (2003) estimated a
25 % increase from the 1950s of global power input to inertial motions.
The subtleties of interior mixing forced by high-frequency winds, as
highlighted by our study, add to the list of challenges awaiting
eddy-permitting/eddy-resolving climate models.
Wind forcing strategy
The Southern Ocean storms are represented as cyclonic Rankine vortices:
τθ=τmaxrRif0≤r<R,τθ=τmaxRrifR≤r,
where τmax is the maximum wind stress, R the radius of the vortex
core (300 km). τθ is set to zero for r>900 km.
τmax is modulated by a sinusoidal seasonal cycle so it varies
from τmax0/2 during austral summer and τmax0
during austral winter, with τmax0=1.5 N m-2. Each
vortex forms and vanishes at the same latitude (no meridional displacement)
but the latitude of formation varies following a Gamma distribution similar
to the meridional distribution inferred from cyclones tracks in Patoux et
al. (2009), with most of the cyclones located between 50 and 70∘ S.
The distribution follows a cycle that repeats each 10 years. The lifetime of
the storms is computed such that one cyclone travels the 2000 km zonal
extension of the domain with full strength (∼2 days). This strategy
leads the storm to wrap around itself during its decaying phase, but note
that this only affects a limited portion of the domain. One storm is formed
every 10 days. The cyclones form and vanish linearly in 1 day, and travel
eastward at a speed Cs of 15 m s-1 in the reference
experiment. Cyclone position and associated winds are recomputed at each time
step.
Rotatory spectra
The computation of rotatory spectra follow the methodology described in
Leaman and Sanford (1975) and others (e.g., Alford et al., 2012). First the
near-inertial part of the velocities uniw are obtained by filtering
the velocity components in the near-inertial band {0.9,1.15}f. These
velocities are then normalized at each depth as follows:
unniw(z)=uniw(z)/N(z)/N0,
where un(z) is the normalized velocity, u(z) is the band-pass
filtered velocity, N(z) is the Brunt–Väisälä frequency and
N0 is the vertical average of N(z). The velocity are then
WKB stretched according to dz′=N(z)/N0dz with z′ the stretched and z the unstretched
coordinates.
Acknowledgements
This study has been supported by CNRS and has been founded by the French ANR
project SMOC. Supercomputing facilities were provided by PRACE project RA1616
and GENCI project GEN1140. The authors wish to thank Y. Cuypers,
E. Pallàs-Sanz, L. Debreu, F. Lemarié and P. Marchesiello for useful
discussions. A special thanks to S. Masson for his assistance in porting NEMO
configurations to Tier0 systems. Interactions with the Communauté de
Modélisation (COMODO, ANR funding) are also acknowledged. We are grateful
to three anonymous reviewers for helpful comments on the
manuscript. Edited by: M. Hecht
References
Abernathey, R., Marshall, J., and Ferreira, D.: The Dependence of
Southern Ocean Meridional Overturning on Wind Stress, J. Phys.
Oceanogr., 41, 2261–2278, 2011.
Alford, M. H.: Improved global maps and 54-year history of wind-work
on ocean inertial motions, Geophys. Res. Lett., 30, 2003.
Alford, M. H., Cronin, M. F., and Klymak, J. M.: Annual Cycle and Depth
Penetration of Wind-Generated Near-Inertial Internal Waves at Ocean Station
Papa in the Northeast Pacific, J. Phys. Oceanogr., 42,
889–909, 2012.
Anderson, D. L. and Gill, A. E.: Beta dispersion of inertial waves,
J. Geophys. Res.-Oceans, 84, 1836–1842, 1979.
Badin, G. and Williams, R. G.: On the buoyancy forcing and residual
circulation in the Southern Ocean: The feedback from Ekman and eddy
transfer, J. Phys. Oceanogr., 40, 295–310, 2010.
Barnier, B., Siefridt, L., and Marchesiello, P.: Thermal forcing for
a global ocean circulation model using a three-year climatology of ECMWF
analyses, J. Marine Syst., 6, 363–380, 1995.
Berbery, E. H. and Vera, C. S.: Characteristics of the Southern
Hemisphere winter storm track with filtered and unfiltered data, J.
Atmos. Sci., 53, 468–481, 1996.
Blaker, A. T., Hirschi, J. J., Sinha, B., De Cuevas, B., Alderson, S.,
Coward, A., and Madec, G.: Large near-inertial oscillations of the
Atlantic meridional overturning circulation, Ocean Model., 42, 50–56, 2012.Cunningham, S. A., Alderson, S. G., King, B. A., and Brandon, M. A.:
Transport and variability of the Antarctic circumpolar current in drake
passage, J. Geophys. Res.-Oceans, 108, 8084, 10.1029/2001JC001147, 2003.Curcic, M., Chen, S. S., and Özgökmen, T. M.: Hurricane-induced
ocean waves and Stokes drift and their impacts on surface transport and
dispersion in the Gulf of Mexico, Geophys. Res. Lett., 43, 2773–2781,
10.1002/2015GL067619, 2016.
Cuypers, Y., Le Vaillant, X., Bouruet-Aubertot, P., Vialard, J., and
Mcphaden, M. J.: Tropical storm-induced near-inertial internal waves during
the Cirene experiment: Energy fluxes and impact on vertical mixing,
J. Geophys. Res.-Oceans, 118, 358–380, 2013.
Danioux, E., Klein, P., and Rivière, P.: Propagation of wind energy
into the deep ocean through a fully turbulent mesoscale eddy field,
J. Phys. Oceanogr., 38, 2224–2241, 2008.
Danioux, E., Klein, P., Hecht, M. W., Komori, N., Roullet, G., and
Le Gentil, S.: Emergence of Wind-Driven Near-Inertial Waves in the Deep Ocean
Triggered by Small-Scale Eddy Vorticity Structures, J. Phys.
Oceanogr., 41, 1297–1307, 2011.de Boyer Montegut, C., Madec, G., Fischer, A. S., Lazar, A., and Iudicone, D.: Mixed layer depth over the global ocean: An examination
of profile data and a profile-based climatology, J. Geophys. Res., 109, C12003,
10.1029/2004JC002378, 2004.De Lavergne C., Madec, G., Le Sommer, J., Nurser, A. G., and
Naveira-Garabato, A. C.: On the consumption of Antarctic Bottom Water in the
abyssal ocean, J. Phys. Oceanogr., 46, 635–651,
10.1175/JPO-D-14-0201.1, 2016.Doglioli, A. M., Blanke, B., Speich, S., and Lapeyre, G.: Tracking
coherent structures in a regional ocean model with wavelet analysis:
Application to Cape Basin eddies, J. Geophys. Res.-Oceans, 112, C05043, 10.1029/2006JC003952, 2007.Elipot, S. and Gille, S. T.: Estimates of wind energy input to the
Ekman layer in the Southern Ocean from surface drifter data, J.
Geophys. Res.-Oceans, 114, C06003, 10.1029/2008JC005170, 2009.
Fox-Kemper, B., Ferrari, R., and Hallberg, R.: Parameterization of
mixed layer eddies – Part I: Theory and diagnosis, J. Phys.
Oceanogr., 38, 1145–1165, 2008.Furuichi, N., Hibiya, T., and Niwa, Y.: Model-predicted distribution of
wind-induced internal wave energy in the world's oceans, J.
Geophys. Res.-Oceans, 113, C09034, 10.1029/2008JC004768, 2008.
Garrett, C.: What is the “near-inertial” band and why is it
different from the rest of the internal wave spectrum?, J. Phys.
Oceanogr., 31, 962–971, 2001.Gent, P. R.: Effects of Southern Hemisphere Wind Changes on the
Meridional Overturning Circulation in Ocean Models, Mar. Sci., 8, 79–94,
10.1146/annurev-marine-122414-033929, 2016.
Gille, S. T., Ledwell, J., Naveira Garabato, A., Speer, K., Balwada, D., Brearley, A., Girton, J. B., Griesel, A., Ferrari, R.,
Klocker, A., LaCasce, J., Lazarevich, P., Mackay, N., Meredith, M. P., Messias, M.-J., Owens, B., Sallée, J.-B., Sheen, K., Shuckburgh, E., Smeed, D. A.,
Laurent, L. C. S., Toole, J. M., Watson, A. J., Wienders, N., and Zajaczkovski, U.: The diapycnal
and isopycnal mixing experiment: a first assessment, CLIVAR Exchanges, 17, 46–48, 2012.
Greatbatch, R. J.: On the response of the ocean to a moving storm: The
nonlinear dynamics, J. Phys. Oceanogr., 13, 357–367, 1983.
Greatbatch, R. J.: On the Response of the Ocean to a Moving Storm:
Parameters and Scales, J. Phys. Oceanogr., 14, 59–78, 1984.
Hogg, A. M., Meredith, M. P., Chambers, D. P., Abrahamsen, E. P., Hughes, C.
W., and Morrison, A. K.: Recent trends in the Southern Ocean eddy
field, J. Geophys. Res.-Oceans, 120, 257–267, 2015.
Ivey, G. N., Winters, K. B., and Koseff, J. R.: Density
stratification, turbulence, but how much mixing?, Annu. Rev. Fluid Mech.,
40, 169–184, 2008.
Jayne, S. R.: The impact of abyssal mixing parameterizations in an
ocean general circulation model, J. Phys. Oceanogr., 39,
1756–1775, 2009.Jochum, M., Briegleb, B. P., Danabasoglu, G., Large, W. G., Norton, N. J.,
Jayne, S. R., Alford, M. H., and Bryan, F. O.: The impact of oceanic
near-inertial waves on climate, J. Climate, 26, 2833–2844, 10.1175/JCLI-D-12-00181.1, 2013.
Joyce, T. M., Toole, J. M., Klein, P., and Thomas, L. N.: A near-inertial
mode observed within a Gulf Stream warm-core ring, J. Geophys.
Res.-Oceans, 118, 1797–1806, 2013.Komori, N., Ohfuchi, W., Taguchi, B., Sasaki, H., and Klein, P.: Deep
ocean inertia-gravity waves simulated in a high-resolution global coupled
atmosphere–ocean GCM, Geophys. Res. Lett., 35, L04610, 10.1029/2007GL032807, 2008.
Kunze, E.: Near-inertial propagation in geostrophic shear, J. Phys.
Oceanogr., 15, 544–565, 1985.
Kunze, E.: The energy-balance in a warm-core rings near-inertial critical layer, J. Phys. Oceanogr., 25,
942–957, 1995.Kurian, J., Colas, F., Capet, X., McWilliams, J. C., and Chelton, D. B.:
Eddy properties in the California current system, J. Geophys.
Res.-Oceans, 116, C08027, 10.1029/2010JC006895, 2011.Leaman, K. D. and Sanford, T. B.: Vertical energy propagation of
inertial waves: A vector spectral analysis of velocity profiles, J.
Geophys. Res., 80, 1975–1978, 10.1029/JC080i015p01975, 1975.
Leclair, M. and Madec, G.: A conservative leap-frog time stepping
method, Ocean Model., 30, 88–94, 2009.
Ledwell, J. R., St. Laurent, L. C., Girton, J. B., and Toole, J. M.:
Diapycnal mixing in the Antarctic circumpolar current, J. Phys.
Oceanogr., 41, 241–246, 2011.
Madec G.: NEMO ocean engine (Draft edition r5171), Note du Pôle
de modélisation, Institut Pierre-Simon Laplace (IPSL), France, No. 27
ISSN No. 1288-1619, 2014.
Marchesiello, P., Capet, X., Menkes, C., and Kennan, S. C.:
Submesoscale dynamics in tropical instability waves, Ocean Model., 39,
31–46, 2011.
Marshall, J. and Radko, T.: Residual-mean solutions for the Antarctic
Circumpolar Current and its associated overturning circulation, J. Phys.
Oceanogr., 33, 2341–2354, 2003.
Marshall, J., Jamous, D., and Nilsson, J.: Reconciling thermodynamic and
dynamic methods of computation of water-mass transformation rates, Deep Sea
Res. Pt. I, 46, 545–572, 1999.
Mazloff, M. R., Ferrari, R., and Schneider, T.: The Force Balance of
the Southern Ocean Meridional Overturning Circulation, J. Phys.
Oceanogr., 43, 1193–1208, 2013.Melet, A., Hallberg, R., Legg, S., and Polzin, K.: Sensitivity of the
Ocean State to the Vertical Distribution of Internal-Tide-Driven Mixing,
J. Phys. Oceanogr., 43, 602–615, 10.1175/JPO-D-12-055.1, 2013.
Morrison, A. K. and Hogg, A. M.: On the relationship between
Southern Ocean overturning and ACC transport, J. Phys.
Oceanogr., 43, 140–148, 2013.Morrison, A. K., Hogg, A. M., and Ward, M. L.: Sensitivity of the
Southern Ocean overturning circulation to surface buoyancy forcing,
Geophys. Res. Lett., 38, L14602, 10.1029/2011GL048031, 2011.Morrow, R., Ward, M. L., Hogg, A. M., and Pasquet, S.: Eddy response to
Southern Ocean climate modes, J. Geophys. Res., 115, C10030,
10.1029/2009JC005894, 2010.MacKinnon, J., Alford, M., Bouruet-Aubertot, P., Bindoff, N., Elipot, S.,
Gille, S., Girton, J., Gregg, M., Hallberg, R., Kunze, E., Naveira Garabato, A., Phillips, H., Pinkel, R.,
Polzin, K., Sanford, T., Simmons, H., and Speer, K.: Using global arrays to investigate
internal-waves and mixing, in: Proceedings of the OceanObs'09 Conference:
Sustained Ocean Observations and Information for Society, Venice, Italy, Venice, Italy, 21–25 September 2009, Vol. 2, 10.5270/OceanObs09.cwp.58, 2009.
Naveira-Garabato, A. C.: A perspective on the future of physical
oceanography, Philos. T. R. Soc. A, 370, 5480–5511, 2012.
Nikurashin, M. and Ferrari, R.: Radiation and dissipation of internal
waves generated by geostrophic flows impinging on small-scale topography:
Application to the Southern Ocean, J. Phys. Oceanogr., 40, 2025–2042, 2010.
Nikurashin, M. and Legg, S.: A mechanism for local dissipation of
internal tides generated at rough topography, J. Phys.
Oceanogr., 41, 378–395, 2011.
Nikurashin, M., Vallis, G. K., and Adcroft, A.: Routes to energy
dissipation for geostrophic flows in the Southern Ocean, Nat. Geosci.,
6, 48–51, 2013.Park, J. J., Kim, K., and Schmitt, R. W.: Global distribution of the
decay timescale of mixed layer inertial motions observed by
satellite-tracked drifters, J. Geophys. Res.-Ocean, 114,
C11010, 10.1029/2008JC005216, 2009.Patoux, J., Yuan, X., and Li, C.: Satellite-based midlatitude cyclone
statistics over the Southern Ocean: 1. Scatterometer-derived pressure fields
and storm tracking, J. Geophys. Res.-Atmos., 114, D04105, 10.1029/2008JD010873, 2009.
Price, J. F.: Upper ocean response to a hurricane, J. Phys.
Oceanogr., 11, 153–175, 1981.
Rath, W., Greatbatch, R. J., and Zhai, X.: Reduction of near-inertial energy
through the dependence of wind stress on the ocean-surface velocity, J.
Geophys. Res.-Oceans, 118, 2761–2773, 2013.Rath, W., Greatbatch, R. J., and Zhai, X.: On the spatial and temporal
distribution of near-inertial energy in the Southern Ocean, J. Geophys.
Res.-Oceans, 119, 359–376, 10.1002/2013JC009246, 2014.Reffray, G., Bourdalle-Badie, R., and Calone, C.: Modelling turbulent
vertical mixing sensitivity using a 1-D version of NEMO, Geosci. Model Dev.,
8, 69–86, 10.5194/gmd-8-69-2015, 2015.Renault, L., Molemaker, M., McWilliams, J., Shchepetkin, A., Lemarié, F.,
Chelton, D., Illig, S., and Hall, A.: Modulation of wind-work by oceanic
current interaction with the atmosphere, J. Phys. Oceanogr., 46, 1685–1704
10.1175/JPO-D-15-0232.1, 2016.
Rintoul, S., Hughes, C., and Olbers, D.: The Antarctic Circumpolar Current
System, in: Ocean Circulation and Climate, edited by: Siedler, G., Church,
J., and Gould, J., Academic Press, New York, 271–302, 2001.Sasaki, H., Klein, P., Qiu, B., and Sasai, Y.: Impact of oceanic-scale
interactions on the seasonal modulation of ocean dynamics by the atmosphere,
Nat. Comm., 5, 10.1038/ncomms6636, 2014.
Sheen, K. L. and Coauthors: Rates and
mechanisms of turbulent dissipation and mixing in the Southern Ocean: Results
from the Diapycnal and Isopycnal Mixing Experiment in the Southern Ocean
(DIMES), J. Geophys. Res.-Oceans, 118, 2774–2792, 2013.
Smith, K. S. and G. K. Vallis: The scales and equilibration of midocean
eddies: Forced-dissipative flow, J. Phys. Oceanogr., 32, 1699–1721, 2002.
Soufflet, Y., Marchesiello, P., Lemarié, F., Jouanno, J., Capet, X.,
Debreu, L., and Benshila, R.: On effective resolution in ocean models, Ocean
Model., 98, 36–50, 2016.
Shchepetkin, A. F. and McWilliams, J. C.: The regional oceanic modeling system (ROMS) – a split-explicit, free-surface,
topography-following-coordinate oceanic model, Ocean Model., 9, 347–404,
2005.
Shih, L. H., Koseff, J. R., Ivey, G. N., andFerziger, J. H.: Parameterization
of turbulent fluxes and scales using homogeneous sheared stably stratified
turbulence simulations, J. Fluid Mech., 525, 193–214, 2005.
Shchepetkin, A. F. and McWilliams, J. C.: The regional oceanic modeling
system (ROMS): a split-explicit, free-surface,
topography-following-coordinate oceanic model, Ocean Model., 9, 347–404,
2005.
Ulbrich, U., Leckebusch, G. C., and Pinto, J. G.: Extra-tropical cyclones in
the present and future climate: a review, Theor. Appl. Climatol., 96,
117–131, 2009.
Vanneste J.: Balance and spontaneous wave generation in geophysical flows,
Ann. Rev. Fluid Mech., 45, 147–172, 2013.Vincent, E. M., Lengaigne, M., Madec, G., Vialard, J., Samson, G., Jourdain,
N. C., Menkes, C. E., and Jullien, S.: Processes setting the characteristics
of sea surface cooling induced by tropical cyclones, J. Geophys. Res., 117,
C02020, 10.1029/2011JC007396, 2012.
Wainer, I., Taschetto, A., Soares, J., de Oliveira, A. P., Otto-Bliesner, B.,
and Brady, E.: Intercomparison of heat fluxes in the South Atlantic – Part
I: the seasonal cycle, J. Climate, 16, 706–714, 2003.
Walin, G.: On the relation between sea surface heat flow and thermal
circulation in the ocean, Tellus, 34, 187–195, 1982.Waterman, S., Naveira Garabato, A. C., and Polzin, K. L.: Internal Waves and
Turbulence in the Antarctic Circumpolar Current, J. Phys. Oceanogr.,
43, 259–282, 2013.
Webb, D. J., de Cuevas, B. A., and Richmond, C. S.: Improved advection
schemes for ocean models, J. Atmos. Ocean. Technol., 15, 1171–1187, 1998.
Wu, L., Jing, Z., Riser, S., and Visbeck, M.: Seasonal and spatial variations
of Southern Ocean diapycnal mixing from Argo profiling floats, Nat. Geosci.,
4, 363–366, 2011.
Wunsch, C.: The work done by the wind on the oceanic general circulation, J.
Phys. Oceanogr., 28, 2332–2340, 1998.Yuan, X., Patoux, J., and Li, C.: Satellite-based midlatitude cyclone
statistics over the Southern Ocean: 2. Tracks and surface fluxes, J. Geophys.
Res.-Atmos., 114, D04106, 10.1029/2008JD010874, 2009.Zhai, X., Greatbatch, R. J., and Sheng, J.: Advective spreading of
storm-induced inertial oscillations in a model of the northwest Atlantic
Ocean, Geophys. Res. Lett., 31, L14315, 10.1029/2004GL020084, 2004.Zhai, X., Greatbatch, R. J., and Sheng, J.: Doppler-Shifted Inertial
Oscillations on a β Plane, J. Phys. Oceanogr., 35, 1480–1488, 2005.Zhai, X., Greatbatch, R. J., and Zhao, J.: Enhanced vertical propagation of
storm-induced near-inertial energy in an eddying ocean channel model,
Geophys. Res. Lett., 32, L18602, 10.1029/2005GL023643, 2005.
Zhai, X., Greatbatch, R. J., Eden, C., and Hibiya, T.: On the loss of
wind-induced near-inertial energy to turbulent mixing in the upper ocean, J.
Phys. Oceanogr., 39, 3040–3045, 2009.