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    <p>Hi,</p>
    <p><br>
    </p>
    <p>I think the reference that David is bringing up is actually also
      useful as a guideline for choosing the Leith and Smagorinsky
      coeficients for the existing implementations.  Fig. 6 in the paper
      compares EKE spectra from a channel simulation using Smagorinsky,
      the traditional (2D) Leith, as well as the generalized "QG Leith".
      The non-dimensional parameters here have all been set to 1. The
      figure shows pretty clearly that with this choice of parameters
      Smagorinsky is overly dissipative, while the standard Leith (i.e.
      Leith_2D in the figure) is not dissipative enough. This is also
      consistent with my experience (and with Smagorinsky’s original
      paper). Griffies and Hallberg (2000) suggest a large value for the
      Smag coefficient (>2.2) based on a conservative upper-limit
      estimate of the grid Reynolds number. If I’m doing it right, the
      same argument would suggest a minimum of about 2.5 for the Leith
      coefficient - larger than what’s typically used (successfully).
      (And I've also run simulations with significantly smaller
      Smagorinsky coefficient stably and happily.)</p>
    <p>So in summary, this seems to suggest viscC[2/4]Leith \gtrsim 1,
      while viscC[2/4]Leith \lesssim 1.<br>
    </p>
    <br>
    <p>To return to Jody's original question about the applicability of
      Leith at higher resolution (and when we care about internal
      waves). I would argue that in this regime neither Leith nor
      Smagorinsky are quite justifiable, so pick your poison…  You could
      maybe even consider to use a little bit of both. If you include
      Smagorinsky (even with a relatively small coefficient) you may be
      able to get away with setting viscC4LeithD=0, since the
      Smagorinsky part should be able to handle a purely divergent mode.
      (The divergence-dependent term in the Leith implementation is
      basically just a hack to get rid of a purely divergent grid-scale
      mode.The latter would be unaffected by the original Leith
      formulation, which is based on 2D non-divergent flow.)<br>
    </p>
    <p><br>
    </p>
    <p>Cheers,</p>
    <p>Malte<br>
    </p>
    <p><br>
    </p>
    <br>
    <div class="moz-cite-prefix">On 4/18/18 9:50 AM, David Ferreira
      wrote:<br>
    </div>
    <blockquote type="cite"
cite="mid:14B4C99DAD42E249A01406B0B97F754B0134BBD1F4@OC11EXPO27.exchange.mit.edu">
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      <div style="direction: ltr;font-family: Tahoma;color:
        #000000;font-size: 10pt;">Hi all,<br>
        May be useful to mention that Fox-Kemper et al. have also
        developed a QG Leith scheme, same idea as 2D Leith but viscosity
        is proportional the QG PV gradient, instead of just the
        vorticity gradient (one may or may not add the divergent part to
        this).<br>
        The claim is that it is slightly more general than the "normal"
        2D Leith scheme<br>
        <br>
        <p><span style="font-size: 12.000000pt; font-family: 'CMR12'">Bachman,
            S. D., Fox-Kemper, B., Pearson, B., 2017. A scale-aware
            subgrid model for quasi-geostrophic turbulence. J. Geophys.
            Res. 122 (2), 1529– 1554.
          </span></p>
        There is another paper in Ocean Modelling.<br>
        <br>
        cheers,<br>
        david<br>
        <br>
        <br>
        <div style="font-family: Times New Roman; color: #000000;
          font-size: 16px">
          <hr tabindex="-1">
          <div style="direction: ltr;" id="divRpF97212"><font size="2"
              face="Tahoma" color="#000000"><b>From:</b> MITgcm-support
              [<a class="moz-txt-link-abbreviated" href="mailto:mitgcm-support-bounces@mitgcm.org">mitgcm-support-bounces@mitgcm.org</a>] on behalf of Jody
              Klymak [<a class="moz-txt-link-abbreviated" href="mailto:jklymak@uvic.ca">jklymak@uvic.ca</a>]<br>
              <b>Sent:</b> Monday, April 16, 2018 10:17 AM<br>
              <b>To:</b> <a class="moz-txt-link-abbreviated" href="mailto:mitgcm-support@mitgcm.org">mitgcm-support@mitgcm.org</a><br>
              <b>Subject:</b> Re: [MITgcm-support] Choosing Leith
              biharmonic co-efficient?<br>
            </font><br>
          </div>
          <div>Hi all,
            <div class=""><br class="">
            </div>
            <div class="">Thanks so much for your advice and in
              particular the Fox-Kemper and Menemenlis reference (which
              Dmitris had given me in another context, so I guess I
              should have read it ;-).</div>
            <div class=""><br class="">
            </div>
            <div class="">Having quickly read it (on a plane, at 5:30
              AM), I’m a little unsure about the applicability of Leith
              once horizontal scales get down to a km or so,
              particularly if one wants to have a reasonable internal
              wave field, there is the statement that the
              divergence-sensing term will negatively impact
              “high-frequency” internal waves.  </div>
            <div class=""><br class="">
            </div>
            <div class="">I will be playing with this the next couple of
              months, but wanted to know the community insights into
              these parameterizations.  Certainly the 10-km spinup expt
              I’m doing now needed Leith biharmonic viscosities to make
              a nice flow.</div>
            <div class=""><br class="">
            </div>
            <div class="">Thanks again,   Jody</div>
            <div class=""><br class="">
            </div>
            <div class=""><br class="">
              <div><br class="">
                <blockquote type="cite" class="">
                  <div class="">On Apr 16, 2018, at  6:27 AM, Ryan
                    Abernathey <<a
                      href="mailto:ryan.abernathey@gmail.com" class=""
                      target="_blank" moz-do-not-send="true">ryan.abernathey@gmail.com</a>>
                    wrote:</div>
                  <br class="Apple-interchange-newline">
                  <div class="">
                    <div dir="ltr" class="">Jody,
                      <div class=""><br class="">
                      </div>
                      <div class="">I assume you are talking about the
                        config you got from Dhruv. The values we use in
                        those runs were taken from Dimitris Menemenlis
                        and Chris Hill's LLC simulations. The exact same
                        settings are used for 1/12, 1/24, and 1/48
                        degree global simulations. (Although I do recall
                        that Dhruv had to make some changes related to
                        boundary conditions.)</div>
                      <div class=""><br class="">
                      </div>
                      <div class="">-Ryan</div>
                    </div>
                    <div class="gmail_extra"><br class="">
                      <div class="gmail_quote">On Mon, Apr 16, 2018 at
                        8:59 AM, Malte Jansen <span dir="ltr" class="">
                          <<a href="mailto:mfj@uchicago.edu" class=""
                            target="_blank" moz-do-not-send="true">mfj@uchicago.edu</a>></span>
                        wrote:<br class="">
                        <blockquote class="gmail_quote" style="margin:0
                          0 0 .8ex; border-left:1px #ccc solid;
                          padding-left:1ex">
                          Just to add on to this: the Leith viscosity is
                          supposed to be "scale aware" (at least within
                          a QG turbulence regime), such that the
                          non-dimensional coefficient should be
                          independent of the grid-scale.<br class="">
                          <br class="">
                          -Malte
                          <div class="HOEnZb">
                            <div class="h5"><br class="">
                              <br class="">
                              <br class="">
                              On 4/16/18 3:16 AM, Martin Losch wrote:<br
                                class="">
                              <blockquote class="gmail_quote"
                                style="margin:0 0 0 .8ex;
                                border-left:1px #ccc solid;
                                padding-left:1ex">
                                Hi Jody,<br class="">
                                <br class="">
                                my reference is
                                pkg/mom_common/mom_calc_visc.F where you
                                can find this:<br class="">
                                <br class="">
                                C     RECOMMENDED VALUES<br class="">
                                C     viscC2Leith=1-3<br class="">
                                C     viscC2LeithD=1-3<br class="">
                                C     viscC4Leith=1-3<br class="">
                                C     viscC4LeithD=1.5-3<br class="">
                                C     viscC2smag=2.2-4 (Griffies and
                                Hallberg,2000)<br class="">
                                C               0.2-0.9
                                (Smagorinsky,1993)<br class="">
                                C     viscC4smag=2.2-4 (Griffies and
                                Hallberg,2000)<br class="">
                                <br class="">
                                And I think this is where it is
                                described:<br class="">
                                B. Fox-Kemper and D. Menemenlis. Can
                                large eddy simulation techniques improve
                                mesoscale-rich ocean models? In M. Hecht
                                and H. Hasumi, editors, Ocean Modeling
                                in an Eddying Regime, volume 177, pages
                                319-338. AGU Geophysical Monograph
                                Series, 2008.<br class="">
                                <a
href="http://www.geo.brown.edu/research/Fox-Kemper/pubs/pdfs/FoxKemperMenemenlis08.pdf"
                                  rel="noreferrer" class=""
                                  target="_blank" moz-do-not-send="true">http://www.geo.brown.edu/resea<wbr
                                    class="">rch/Fox-Kemper/pubs/pdfs/FoxKe<wbr
                                    class="">mperMenemenlis08.pdf</a><br
                                  class="">
                                <br class="">
                                Martin<br class="">
                                <br class="">
                                <blockquote class="gmail_quote"
                                  style="margin:0 0 0 .8ex;
                                  border-left:1px #ccc solid;
                                  padding-left:1ex">
                                  On 14. Apr 2018, at 22:42, Jody Klymak
                                  <<a href="mailto:jklymak@uvic.ca"
                                    class="" target="_blank"
                                    moz-do-not-send="true">jklymak@uvic.ca</a>>
                                  wrote:<br class="">
                                  <br class="">
                                  <br class="">
                                  Hi all,<br class="">
                                  <br class="">
                                  Is there a good reference and/or ruls
                                  of thumb for choosing the Leith
                                  biharmonic co-efficient? i.e.
                                  `viscC4Leith` and `viscC4Leithd`, with
                                  particular interest in how the size of
                                  the grid affects the choice?  I’m
                                  using someone else’s setup, using
                                  Leith works great for making the flow
                                  look reasonable, but it would be nice
                                  if it was turned down as much as
                                  possible, particularly when I
                                  downscale to a smaller grid size…<br
                                    class="">
                                  <br class="">
                                  Thanks a lot,   Jody<br class="">
                                  ______________________________<wbr
                                    class="">_________________<br
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                              <br class="">
                            </div>
                          </div>
                          <span class="HOEnZb"><font class=""
                              color="#888888">-- <br class="">
                              Malte F Jansen<br class="">
                              Assistant Professor<br class="">
                              Department of the Geophysical Sciences<br
                                class="">
                              The University of Chicago<br class="">
                              5734 South Ellis Avenue<br class="">
                              Chicago, IL 60637 USA</font></span>
                          <div class="HOEnZb">
                            <div class="h5"><br class="">
                              <br class="">
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    <br>
    <pre class="moz-signature" cols="72">-- 
Malte F Jansen
Assistant Professor
Department of the Geophysical Sciences
The University of Chicago
5734 South Ellis Avenue
Chicago, IL 60637 USA</pre>
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