_Governing.dox

1/*! \page Governing_Equations Governing Equations

2

3MOM6 is a hydrostatic ocean circulation model that time steps either

4the non-Boussinesq ocean equations (where the flow velocity is

5divergent: \f$\nabla \cdot \mathbf{v} \ne 0\f$), or the Boussinesq

6ocean equations (where velocity is non-divergent: \f$\nabla \cdot

7\mathbf{v} = 0\f$). We here display the Boussinesq version since

8it is most commonly used (as of 2022). We start by casting the

9equations in geopotentiial coordinates prior to transforming to the

10generalized vertical coordinates used by MOM6. A more thorough

11discussion of these equations, and their finite volume realization

12appropriate for MOM6, can be found in Griffies, Adcroft and Hallberg (2020)

13\cite Griffies_Adcroft_Hallberg2020.

14

15The hydrostatic Boussinesq ocean equations, written using geopotential

16vertical coordinates, are given by

17\f{align}

18 \rho_o \left[

19 D_t \mathbf{u} + f \hat{\mathbf{z}} \times \mathbf{u}

20 \right]

21 &= -\rho \, \nabla_z \Phi - \nabla_z p

22 + \rho_o \, \mathbf{\mathcal{F}}

23 &\mbox{horizontal momentum}

24\\

25 \rho \, \partial_{z} \Phi + \partial_{z} p &= 0 &\mbox{hydrostatic}

26\\

27 \nabla_z \cdotp \mathbf{u} + \partial_{z} w

28 &= 0

29 &\mbox{continuity}

30\\

31 D_t \theta &= \mathbf{\mathcal{N}}_\theta^\gamma

32 - \partial_{z} J_\theta^{(z)}

33 &\mbox{potential or Conservative temp}

34 \\

35 D_t S &= \mathbf{\mathcal{N}}_S^\gamma

36- \partial_{z} J_S^{(z)}

37 &\mbox{salinity}

38\\

39 \rho &= \rho(S, \theta, z) &\mbox{ equation of state}

40\\

41 \mathbf{v} &= \mathbf{u} + \hat{\mathbf{z}} \, w &\mbox{velocity field.}

42\f}

43

44The acceleration term, \f$\mathbf{\mathcal{F}}\f$, in the

45horizontal momentum equation includes the acceleration due to the

46divergence of internal frictional stresses as well as from bottom and

47surface boundary stresses. Other notation is described in \ref

48Notation.

49

50The prognostic temperature, \f$\theta\f$, is either potential

51temperature or Conservative Temperature, depending on the chosen

52equation of state, and \f$S\f$ is the salinity. We generally follow

53the discussion of \cite McDougall_etal_2021 for how to interpret the

54prognostic temperature and salinity in ocean models. MOM6 has

55historically used the Wright (1997) \cite wright1997 equation of state

56to compute the <em>in situ</em> density, \f$\rho\f$. However, there

57are other options as documented in \ref Equation_of_State. In the

58potential temperature and salinity equations, fluxes due to diabatic

59processes are indicated by \f$J^{(z)}\f$. Tendencies due to the

60convergence of fluxes oriented along neutral directions are indicated

61by \f$\mathbf{\mathcal{N}}^\gamma\f$, with our implementation of

62<em>neutral diffusion</em> detailed in Shao et al (2020)

63\cite Shao_etal_2020.

64

65The total or material time derivative operator is given by

66\f{align}

67 D_t &\equiv \partial_{t} + \mathbf{v} \cdotp \nabla

68 \\

69 &= \partial_{t} + \mathbf{u} \cdotp \nabla_z + w \, \partial_{z},

70\f}

71where the second equality explosed the horizontal and vertical terms. Using the non-divergence condition

72on the three-dimensional velocity allows us to write the material time derivative of an arbitrary scalar field,

73\f$\psi\f$, into a flux-form equation

74\f{align} D_t \psi &= ( \partial_{t} + \mathbf{u} \cdotp \nabla) \, \psi

75 \\

76 &= \partial_{t} \psi + \nabla \cdotp (\mathbf{v} \, \psi)

77\\

78 &= \partial_{t} \psi + \nabla_z \cdotp ( \mathbf{u} \, \psi) + \partial_{z} ( w \, \psi).

79\f}

80Discretizing the flux-form scalar equations means that fluxes

81transferring scalars between grid cells act in a conservative manner.

82Consequently, the domain integrated scalar (e.g., total seawater volume, total

83salt content, total potential enthalpy) is affected only via surface and bottom

84boundary transport. Such global conservation properties are

85maintained by MOM6 to within computational roundoff, with this level

86of precision found to be essential for using MOM6 to study

87climate. Making use of the flux-form scalar conservation equations

88brings the model equations to the form

89\f{align}

90 \rho_o \left[

91 D_t \mathbf{u} + f \hat{\mathbf{z}} \times \mathbf{u}

92 \right]

93 &= -\rho \, \nabla_z \Phi - \nabla_z p

94 + \rho_o \, \mathbf{\mathcal{F}}

95 &\mbox{horizontal momentum}

96\\

97 \rho \, \partial_{z} \Phi + \partial_{z} p &= 0 &\mbox{hydrostatic}

98\\

99 \nabla_z \cdotp \mathbf{u} + \partial_{z} w

100 &= 0

101 &\mbox{continuity}

102\\

103\partial_{t} \theta + \nabla_z \cdotp (\mathbf{u} \, \theta) + \partial_{z} (w \, \theta)

104&= \mathbf{\mathcal{N}}_\theta^\gamma - \partial_{z} J_\theta^{(z)}

105&\mbox{potential or Conservative temp}

106\\

107\partial_{t} S + \nabla_z \cdotp (\mathbf{u} \, S) + \partial_{z}(w \, S)

108&= \mathbf{\mathcal{N}}_S^\gamma -\partial_{z} J_S^{(z)}

109 &\mbox{salinity}

110\\

111\rho &= \rho(S, \theta, z) &\mbox{equation of state.}

112\f}

113

114\section vector_invariant_eqns Vector invariant velocity equation

115

116MOM6 time steps the horizontal velocity equation in its

117vector-invariant form. To derive this equation we make use of the

118following vector identity

119\f{align}

120 D_t \mathbf{u}

121 &=

122 \partial_t \mathbf{u} + \mathbf{v} \cdotp \nabla \mathbf{u}

123 \\

124 &=

125 \partial_t \mathbf{u} + \mathbf{u} \cdotp \nabla_z \mathbf{u} + w \partial_z \mathbf{u}

126 \\

127 &=

128 \partial_t \mathbf{u} + \left( \nabla \times \mathbf{u} \right) \times \mathbf{v}

129 + \nabla \left|\mathbf{u}\right|^2/2

130 \\

131 &=

132 \partial_t \mathbf{u} + w \, \partial_{z} \mathbf{u}

133 + \zeta \, \hat{\mathbf{z}} \times \mathbf{u} + \nabla_{z} K,

134\f}

135where we introduced the vertical component to the relative vorticity

136\f{align}

137 \zeta = \hat{\mathbf{z}} \cdot (\nabla \times \mathbf{u})

138 = \partial_{x}v - \partial_{y} u,

139\label{eq:relative-vorticity-z}

140\f}

141as well as the kinetic energy per mass contained in the horizontal flow

142\f{align}

143 K = (u^{2} + v^{2})/2.

144\label{eq:kinetic-energy-per-mass}

145\f}

146It is just the horizontal kinetic energy per mass that appears when

147making the hydrostatic approximation, whereas a non-hydrostatic fluid

148(such as the MITgcm) includes the contribution from vertical motion. With

149these identities we are led to the MOM6 flux-form equations of motion in

150geopotential coordinates

151\f{align}

152 \rho_{o} \left[

153 \partial_t \mathbf{u} + w \, \partial_{z} \mathbf{u}

154 + (f + \zeta) \hat{\mathbf{z}} \times \mathbf{u}

155 \right]

156 &= -\nabla_{z} (p + K) - \rho \, \nabla_{z} \Phi + \rho_{o} \, \mathbf{\mathcal{F}}

157 &\mbox{vector-inv horz velocity}

158\\

159 \rho \, \partial_{z} \Phi + \partial_{z} p &= 0 &\mbox{hydrostatic}

160\\

161 \nabla_z \cdotp \mathbf{u} + \partial_{z} w

162 &= 0

163 &\mbox{continuity}

164 \\

165 \partial_t \theta + \nabla_z \cdotp ( \mathbf{u} \, \theta ) + \partial_z ( w \, \theta )

166 &= \mathbf{\mathcal{N}}_\theta^\gamma - \partial_{z} J_\theta^{(z)}

167 &\mbox{potential/Cons temp}

168 \\

169 \partial_t S + \nabla_z \cdotp ( \mathbf{u} \, S ) + \partial_z (w \, S)

170 &= \mathbf{\mathcal{N}}_S^\gamma - \partial_{z} J_S^{(z)}

171 &\mbox{salinity}

172 \\

173 \rho &= \rho(S, \theta, z) &\mbox{equation of state.}

174\f}

175

176*/