.. role:: cpp(code)
    :language: c++

.. _Climatology:


Nudging to climatology
======================

REMORA supports nudging to climatology both on the boundaries and in an interior sponge region. Nudging at the boundaries is discussed in :ref:`Boundary Conditions<sec:domainBCs>`.

The climatology history and nudging coefficients are read from NetCDF files. The climatology history may come from a single file or a time-ordered list of files. The nudging coefficients are expected to be in units of 1/day, as in ROMS.

For AMR runs, climatology fields are read and temporally interpolated on level 0, then interpolated to higher AMR levels as needed. This allows one climatology dataset on the coarse grid to drive nudging on all levels.

Nudging for Tracers
-------------------

Tracers :math:`S`  are nudged towards solution by:

.. math::
  S = S + C_{\mathrm{tracer}} \Delta t \left(S_{\mathrm{clim}} - S\right)

where :math:`S_{mathrm{clim}}` is the value given in the climatology and :math:`C_{\mathrm{tracer}}` is the nudging coefficient for the relevant tracer.

Nudging for 2D velocity
-----------------------

The 2D velocity components are updated in the right-hand side term ``rhs_ubar`` according to:

.. math::
   \mathrm{RHS}_{\overline{u}} = \mathrm{RHS}_{\overline{u}} + C_{\mathrm{M2}} \frac{D}{mn} \left(\overline{u}_{\mathrm{clim}} - \overline{u}_{\mathrm{krhs}}\right)

or similar for v. :math:`C_{\mathrm{M2}}` is the nudging coefficient, :math:`D` is the water column depth, and :math:`\overline{u}_{\mathrm{clim}}` is the climatology value read from file. :math:`\mathrm{RHS}_{\overline{u}}` is the variable ``rhs_ubar`` and :math:`\overline{u}_{\mathrm{krhs}}` is the value of :math:`\overline{u}` at the time index ``krhs``.

Nudging for 3D velocity
-----------------------

The 3D velocity components are updated in the right-hand side term ``ru`` according to:

.. math::
   \mathrm{RHS}_{u} = \mathrm{RHS}_{u} + C_{\mathrm{M3}} \frac{H_{z}}{mn} \left(u_{\mathrm{clim}} - u_{\mathrm{old}}\right)

or similar for v. :math:`C_{\mathrm{M3}}` is the nudging coefficient and :math:`u_{\mathrm{clim}}` is the climatology value read from file. :math:`\mathrm{RHS}_{u}` is the variable ``ru`` and :math:`u_{\mathrm{old}}` is the value of :math:`u` at the prior time step.