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8679f9097b Jeff* .. _sub_phys_pkg_kl10:
                 
                 KL10: Vertical Mixing Due to Breaking Internal Waves
                 ----------------------------------------------------
                 
                 
                 (in directory: *pkg/kl10/*)
                 
                 Authors: Jody M. Klymak
                 
                 .. _ssub_phys_pkg_kl10_intro:
                 
                 Introduction
                 ++++++++++++
                 
                 The :cite:`klymaklegg10` parameterization for breaking internal waves is meant to represent
                
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 mixing in the ocean “interior” due to convective instability. Many
                 mixing schemes in the presence of unstable stratification simply turn on
                 an arbitrarily large diffusivity and viscosity in the overturning
                 region. This assumes the fluid completely mixes, which is probably not a
                 terrible assumption, but it also makes estimating the turbulence
                 dissipation rate in the overturning region meaningless.
                 
                 The KL10 scheme overcomes this limitation by estimating the viscosity
                 and diffusivity from a combination of the Ozmidov relation and the
                 Osborn relation, assuming a turbulent Prandtl number of one. The Ozmidov
                 relation says that outer scale of turbulence in an overturn will scale
                 with the strength of the turbulence :math:`\epsilon`, and the
                 stratification :math:`N`, as
                 
                 .. math::
                    :label: eq-pkg-kl10-Lo
                 
                    L_O^2 \approx \epsilon N^{-3}.
                 
                 The Osborn relation relates the strength of the dissipation to the
                 vertical diffusivity as
                 
                 .. math:: K_{v}=\Gamma \epsilon N^{-2},
                 
                 where :math:`\Gamma\approx 0.2` is the mixing ratio of buoyancy flux to
                 thermal dissipation due to the turbulence. Combining the two gives us
                 
                 .. math:: K_{v} \approx \Gamma L_O^2 N.
                 
                 The ocean turbulence community often approximates the Ozmidov scale by
                 the root-mean-square of the Thorpe displacement, :math:`\delta_z`, in an
                 overturn :cite:`thorpe77`. The Thorpe displacement is the distance one would have to
                 move a water parcel for the water column to be stable, and is readily
                 measured in a measured profile by sorting the profile and tracking how
                 far each parcel moves during the sorting procedure. This method gives an
                 imperfect estimate of the turbulence, but it has been found to agree on
                 average over a large range of overturns :cite:`wesson94,seimgregg94,moum96`.
                 
                 The algorithm coded here is a slight simplification of the usual Thorpe
                 method for estimating turbulence in overturning regions. Usually,
                 overturns are identified and :math:`N` is averaged over the overturn.
                 Here, instead we estimate
                 
                 .. math:: K_{v}(z) \approx \Gamma \delta_z^2\, N_s(z).
                 
                 where :math:`N_s(z)` is the local sorted stratification. This saves
                
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 complexity in the code and adds a slight inaccuracy, but we don’t
                 believe is biased.
                 
                 We assume a turbulent Prandtl number of 1, so :math:`A_v=K_{v}`.
                 
                 We also calculate and output a turbulent dissipation from this scheme.
                 We do not simply evaluate the overturns for :math:`\epsilon` using
8586b5df8e Mart* :eq:`eq-pkg-kl10-Lo`. Instead we compute the vertical shear terms that the
8679f9097b Jeff* viscosity is acting on:
                 
0bad585a21 Navi* .. math:: \epsilon_v = A_v \left[ \left(\partial_z u \right)^2 + \left( \frac{\partial u}{\partial z} \right)^2 \right].
8679f9097b Jeff* 
                 There are straightforward caveats to this approach, covered in :cite:`klymaklegg10`.
                 
                 -  If your resolution is too low to resolve the breaking internal waves,
                
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    you won’t have any turbulence.
                 
                 -  If the model resolution is too high, the estimates of
                    :math:`\epsilon_v` will start to be exaggerated, particularly if the
                    run in non-hydrostatic. That is because there will be significant
                    shear at small scales that represents the turbulence being
                    parameterized in the scheme. At very high resolutions direct
                    numerical simulation or more sophisticated large-eddy schemes should
                    be used.
                 
                 -  We find that grid cells of approximately 10 to 1 aspect ratio are a
                    good rule of thumb for achieving good results are usual oceanic
                    scales. For a site like the Hawaiian Ridge, and Luzon Strait, this
                    means 10-m vertical resolusion and approximately 100-m horizontal.
                    The 10-m resolution can be relaxed if the stratification drops, and
                    we often WKB-stretch the grid spacing with depth.
                 
                 -  The dissipation estimate is useful for pinpoiting the location of
                    turbulence, but again, is grid size dependent to some extent, and
                    should be treated with a grain of salt. It will also not include any
                    numerical dissipation such as you may find with higher order
                    advection schemes.
                 
                 
                 .. _ssub_phys_pkg_kl10_comp:
                 
8586b5df8e Mart* KL10 configuration and compiling
8679f9097b Jeff* ++++++++++++++++++++++++++++++++
                 
                 As with all MITgcm packages, KL10 can be turned on or off at compile
                 time
                 
                 -  using the ``packages.conf`` file by adding ``kl10`` to it,
                 
                 -  or using ``genmake2`` adding ``-enable=kl10`` or ``-disable=kl10``
                    switches
                 
                 -  *Required packages and CPP options:*
                    No additional packages are required.
                 
                 (see Section [sec:buildingCode]).
                 
                 KL10 has no compile-time options (``KL10_OPTIONS.h`` is empty).
                 
                 
                 .. _ssub_phys_pkg_kl10_runtime:
                 
                 Run-time parameters
                 +++++++++++++++++++
                 
                 Run-time parameters are set in files ``data.pkg`` and ``data.kl10``
                 which are read in ``kl10_readparms.F``. Run-time parameters may be
                 broken into 3 categories: (i) switching on/off the package at runtime,
                 (ii) required MITgcm flags, (iii) package flags and parameters.
                 
                 Enabling the package
                 ####################
                 
                 The KL10 package is switched on at runtime by setting
                 ``useKL10 = .TRUE.`` in ``data.pkg``.
                 
                 Required MITgcm flags
                 #####################
                 
                 The following flags/parameters of the MITgcm dynamical kernel need to
                 be set in conjunction with KL10:
                 
                 +----------------------------------+--------------------------------------+
                 | ``implicitViscosity = .TRUE.``   | enable implicit vertical viscosity   |
                 +----------------------------------+--------------------------------------+
                 | ``implicitDiffusion = .TRUE.``   | enable implicit vertical diffusion   |
                 +----------------------------------+--------------------------------------+
                 
                 Package flags and parameters
                 ############################
                 
                 :numref:`tab_phys_pkg_kl10_runtime` summarizes the runtime
                 flags that are set in ``data.kl10``, and their default values.
                 
                 
                 .. table:: KL10 runtime parameters.
                   :name: tab_phys_pkg_kl10_runtime
                 
                   +----------------------+---------------------------------+----------------------------------------------+
                   | **Flag/parameter**   | **default**                     | **Description**                              |
                   +======================+=================================+==============================================+
                   | KLviscMax            | 300�\ m\ :sup:`2` s\ :sup:`--1` | Maximum viscosity the scheme will ever give  |
                   |                      |                                 | (useful for stability)                       |
                   +----------------------+---------------------------------+----------------------------------------------+
                   | KLdumpFreq           | ``dumpFreq``                    | Dump frequency of KL10 field snapshots       |
                   +----------------------+---------------------------------+----------------------------------------------+
                   | KLtaveFreq           | ``taveFreq``                    | Averaging and dump frequency of KL10 fields  |
                   +----------------------+---------------------------------+----------------------------------------------+
                   | KLwriteState         | ``.FALSE.``                     | write KL10 state to file                     |
                   +----------------------+---------------------------------+----------------------------------------------+
                 
                 .. _ssub_phys_pkg_kl10_equations:
                 
                 Equations and key routines
                 ++++++++++++++++++++++++++
                 
                 KL10_CALC:
                 ###########
                 
                 Top-level routine. Calculates viscosity and diffusivity on the grid cell
                 centers. Note that the runtime parameters ``viscAz`` and ``diffKzT`` act
                 as minimum viscosity and diffusivities. So if there are no overturns (or
                 they are weak) then these will be returned.
                 
                 KL10_CALC_VISC:
                 ###############
                 
                 Calculates viscosity on the W and S grid faces for U and V respectively.
                 
                 KL10_CALC_DIFF:
                 ###############
                 
                 Calculates the added diffusion from KL10.
                 
                 .. _ssub_phys_pkg_kl10_diagnostics:
                 
                 KL10 diagnostics
                 ++++++++++++++++
                 
                 Diagnostics output is available via the diagnostics package (see Section
                 [sec:pkg:diagnostics]). Available output fields are summarized here:
                 
                 ::
                 
                     ------------------------------------------------------
                      <-Name->|Levs|grid|<--  Units   -->|<- Tile (max=80c)
                     ------------------------------------------------------
                      KLviscAr| Nr |SM  |m^2/s           |KL10 vertical eddy viscosity coefficient
                      KLdiffKr| Nr |SM  |m^2/s           |Vertical diffusion coefficient for salt, temperature, & tracers
                      KLeps   | Nr |SM  |m^3/s^3         |Turbulence dissipation estimate.
                 
                 
                 .. _ssub_phys_pkg_kl10_examples:
                 
                 
                 References
                 ++++++++++
                 
                 Klymak and Legg, 2010, *Oc.�Modell.*.
                 
                 
                 Experiments and tutorials that use KL10
                 +++++++++++++++++++++++++++++++++++++++
                 
                 -  Modified Internal Wave experiment, in internal_wave verification
                    directory

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