doc/overview/soln_strategy.rst

f67abf1ee3 Jeff*0001 Solution strategy
                0002 -----------------
                0003
                0004 The method of solution employed in the **HPE**, **QH** and **NH** models
                0005 is summarized in :numref:`soln_strat`. Under all dynamics, a
                0006 2-d elliptic equation is first solved to find the surface pressure and
                0007 the hydrostatic pressure at any level computed from the weight of fluid
                0008 above. Under **HPE** and **QH** dynamics, the horizontal momentum
                0009 equations are then stepped forward and :math:`\dot{r}` found from
                0010 continuity. Under **NH** dynamics a 3-d elliptic equation must be solved
                0011 for the non-hydrostatic pressure before stepping forward the horizontal
                0012 momentum equations; :math:`\dot{r}` is found by stepping forward the
                0013 vertical momentum equation.
                0014
                0015 There is no penalty in implementing **QH** over **HPE** except, of
                0016 course, some complication that goes with the inclusion of
                0017 :math:`\cos \varphi \ ` Coriolis terms and the relaxation of the shallow
                0018 atmosphere approximation. But this leads to negligible increase in
                0019 computation. In **NH**, in contrast, one additional elliptic equation -
0bad585a21 Navi*0020 a three-dimensional one - must be inverted for :math:`p_{\rm nh}`. However
f67abf1ee3 Jeff*0021 the ‘overhead’ of the **NH** model is essentially negligible in the
                0022 hydrostatic limit (see detailed discussion in Marshall et al. (1997) :cite:`marshall:97a`
                0023 resulting in a non-hydrostatic algorithm that, in the hydrostatic limit,
                0024 is as computationally economic as the **HPEs**.
                0025
                0026
                0027
                0028   .. figure:: figs/solution_strategy.*
                0029     :width: 100%
                0030     :align: center
                0031     :alt: diagram of basic solution strategy in MITgcm
                0032     :name: soln_strat
                0033
                0034     Basic solution strategy in MITgcm. **HPE** and **QH** forms diagnose the vertical velocity, in **NH** a prognostic equation for the vertical velocity is integrated.