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f67abf1ee3 Jeff*0001 Forcing/dissipation
0002 -------------------
0003
0004 Forcing
0005 ~~~~~~~
0006
0007 The forcing terms :math:`\mathcal{F}` on the rhs of the equations are
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0008 provided by ‘physics packages’ and forcing packages. These are described
0009 later on.
0010
0011 Dissipation
0012 ~~~~~~~~~~~
0013
0014 Momentum
0015 ^^^^^^^^
0016
0017 Many forms of momentum dissipation are available in the model. Laplacian
0018 and biharmonic frictions are commonly used:
0019
0020 .. math::
0021 D_{V}=A_{h}\nabla _{h}^{2}v+A_{v}\frac{\partial ^{2}v}{\partial z^{2}}
0022 +A_{4}\nabla _{h}^{4}v
0023 :label: dissipation
0024
0025 where :math:`A_{h}` and :math:`A_{v}\ `\ are (constant) horizontal and
0026 vertical viscosity coefficients and :math:`A_{4}\ `\ is the horizontal
0027 coefficient for biharmonic friction. These coefficients are the same for
0028 all velocity components.
0029
0030 Tracers
0031 ^^^^^^^
0032
0033 The mixing terms for the temperature and salinity equations have a
0034 similar form to that of momentum except that the diffusion tensor can be
0035 non-diagonal and have varying coefficients.
0036
0037 .. math::
0bad585a21 Navi*0038 D_{T,S} = \nabla \cdot \left[ \boldsymbol{K} \nabla (T,S) \right] + K_{4} \nabla
0039 _{h}^{4}(T,S),
f67abf1ee3 Jeff*0040 :label: diffusion
0041
0bad585a21 Navi*0042 where :math:`\boldsymbol{K}` is the diffusion tensor and
0043 :math:`K_{4}\ ` the horizontal coefficient for biharmonic diffusion. In
f67abf1ee3 Jeff*0044 the simplest case where the subgrid-scale fluxes of heat and salt are
0045 parameterized with constant horizontal and vertical diffusion
0bad585a21 Navi*0046 coefficients, :math:`\boldsymbol{K}`, reduces to a diagonal
f67abf1ee3 Jeff*0047 matrix with constant coefficients:
0048
0049 .. math::
0bad585a21 Navi*0050 \qquad \qquad \qquad \qquad \boldsymbol{K} = \left(
f67abf1ee3 Jeff*0051 \begin{array}{ccc}
0052 K_{h} & 0 & 0 \\
0053 0 & K_{h} & 0 \\
0054 0 & 0 & K_{v}
0055 \end{array}
0056 \right) \qquad \qquad \qquad
0057 :label: diagonal-diffusion-tensor
0058
0059 where :math:`K_{h}\ `\ and :math:`K_{v}\ `\ are the horizontal and
0060 vertical diffusion coefficients. These coefficients are the same for all
0061 tracers (temperature, salinity ... ).
0062
