_General_coordinate.dox

1/*! \page General_Coordinate Generalized vertical coordinate equations

2

3The ocean equations discretized by MOM6 are formulated using

4generalized vertical coordinates. Motivation for using generalized

5vertical coordinates, and a full accounting of the ocean equations

6written using these coordinates, can be found in Griffies, Adcroft and

7Hallberg (2020) \cite Griffies_Adcroft_Hallberg2020. Here we provide

8a brief summary.

9

10Consider a smooth function of space and time, \f$r(x,y,z,t)\f$, that

11has a single-signed and non-zero vertical derivative known as the

12specific thickness

13\f{align}

14 \partial z/\partial r = (\partial r/\partial z)^{-1} = \mbox{specific thickness.}

15\f}

16The specific thickness measures the inverse vertical stratification of

17the vertical coordinate surfaces. As so constrained, \f$r\f$ can

18uniquely prescribe a positiion in the vertical. Consequently, the

19ocean equations can be mapped one-to-one from geopotential vertical

20coordinates to generalized vertical coordinate. Upon transforming to

21\f$r\f$-coordinates, the material time derivative of \f$r\f$ appears

22throughout the equations, playing the role of a pseudo-vertical

23velocity, and we make use of the following shorthand for this

24derivative

25\f{align}

26\dot{r} = D_{t} r.

27\f}

28

29The Boussinesq hydrostatic ocean equations take the following form using

30generalized vertical coordinates (\f$r\f$-coordinates)

31\f{align}

32\label{html:r-equations}\notag \\

33\rho_o \left[

34 \partial_{t} \mathbf{u} + (f + \zeta) \, \hat{\mathbf{z}} \times \mathbf{u}

35 + \dot{r} \, \partial_{r} \mathbf{u} \right]

36 &= -\nabla_r \, (p + \rho_{o} \, K) -\rho \nabla_r \Phi + \rho_{o} \, \mathbf{\mathcal{F}}

37 &\mbox{horizontal momentum}

38\label{eq:r-horz-momentum}

39\\

40\rho \, \partial_{r} \Phi + \partial_{r}p

41 &= 0

42&\mbox{hydrostatic}

43\label{eq:r-hydrostatic-equation}

44\\

45 \partial_{t}( z_r)

46+ \nabla_r \cdot ( z_r \, \mathbf{u} )

47+ \partial_{r} ( z_r \, \dot{r} )

48&= 0

49&\mbox{specific thickness}

50\label{eq:r-non-divergence}

51\\

52 \partial_{t} ( \theta \, z_r )

53+ \nabla_r \cdot ( \theta z_r \, \mathbf{u} )

54+ \partial_{r} ( \theta \, z_r \, \dot{r} )

55&= z_r \mathbf{\mathcal{N}}_\theta^\gamma

56- \partial_{r} J_\theta^{(z)}

57&\mbox{potential/Conservative temp}

58\label{eq:r-temperature-equation}

59\\

60\partial_{t} ( S \, z_r)

61+ \nabla_r \cdot ( S \, z_r \, \mathbf{u} )

62+ \partial_{r} ( S \, z_r \, \dot{r} )

63&= z_r \mathbf{\mathcal{N}}_S^\gamma

64- \partial_{r} J_S^{(z)}

65&\mbox{salinity}

66\label{eq:r-salinity-equation}

67\\

68\rho &= \rho( S, \theta, -g \rho_0 z )

69&\mbox{equation of state.}

70\f}

71The time derivatives appearing in these equations are computed with

72the generalized vertical coordinate fixed rather than the

73geopotential. It is a common misconception that the horizontal

74velocity, \f$\mathbf{u}\f$, is rotated to align with constant \f$r\f$

75surfaces. Such is not the case. Rather, the horizontal velocity,

76\f$\mathbf{u}\f$, is precisely the same horizontal velocity used with

77geopotential coordinates. However, its evolution has here been

78formulated using generalized vertical coordinates.

79

80As a finite volume model, MOM6 is discretized in the vertical by

81integrating between surfaces of constant \f$r\f$. The layer thickness

82is a basic term appearing in these equations, which results from

83integrating the specific thickness over a layer

84\f{align}

85h = \int z_r \, \mathrm{d}r.

86\f}

87Correspondingly, the model variables are treated as finite volume

88averages over each layer, with full accounting of this finite volume

89approach presented in Griffies, Adcroft and Hallberg (2020)

90\cite Griffies_Adcroft_Hallberg2020, and with the semi-discrete model

91ocean model equations written as follows.

92\f{align}

93\rho_0

94\left[ \frac{\partial \mathbf{u}}{\partial t} + \frac{( f + \zeta )}{h} \,

95\hat{\mathbf{z}} \times h \, \mathbf{u} + \underbrace{ \dot{r} \,

96\frac{\partial \mathbf{u}}{\partial r} }

97\right]

98&= -\nabla_r \, (p + \rho_{0} \, K) -

99\rho \nabla_r \, \Phi + \mathbf{\mathcal{F}}

100&\mbox{horizontal momentum}

101\label{eq:h-horz-momentum}

102\\

103\rho \, \delta_r \Phi + \delta_r p

104&= 0

105&\mbox{hydrostatic}

106\label{eq:h-hydrostatic-equation}

107\\

108\frac{\partial h}{\partial t} + \nabla_r \cdot \left( h \, \mathbf{u} \right) +

109\underbrace{ \delta_r ( z_r \dot{r} ) }

110 &= 0

111&\mbox{thickness}

112\label{eq:h-thickness-equation}

113\\

114\frac{\partial ( \theta \, h )}{\partial t} + \nabla_r \cdot \left( \theta h \,

115\mathbf{u} \right) + \underbrace{ \delta_r ( \theta \, z_r \dot{r} ) }

116&=

117h \mathbf{\mathcal{N}}_\theta^\gamma - \delta_r J_\theta^{(z)}

118&\mbox{potential/Conservative temp}

119\label{eq:h-temperature-equation}

120\\

121\frac{\partial ( S \, h )}{\partial t} + \nabla_r \cdot \left( S \, h \,

122\mathbf{u} \right) + \underbrace{ \delta_r ( S \, z_r \dot{r} ) }

123&=

124h \mathbf{\mathcal{N}}_S^\gamma - \delta_r J_S^{(z)}

125&\mbox{salinity}

126\label{eq:h-salinity-equation}

127\\

128\rho &= \rho\left( S, \theta, -g \rho_0 z(r) \right)

129&\mbox{equation of state,} \label{eq:h-equation-of-state}

130\f}

131where

132\f{align}

133\delta_{r} = \mathrm{d}r \, (\partial/\partial r)

134\f}

135is the discrete vertical difference operator. The pressure gradient

136accelerations in the momentum equation are written in

137continuous-in-the-vertical form for brevity; the exact discretization

138is detailed in \cite adcroft2008 and

139\cite Griffies_Adcroft_Hallberg2020. The \f$1/h\f$ and \f$h\f$ appearing in

140the horizontal momentum equation are carefully handled in the code to

141ensure proper cancellation even when the layer thickness goes to zero

142i.e., l'Hospital's rule is respected.

143

144The MOM6 time-stepping algorithm integrates the above layer-averaged

145equations forward in time allowing the vertical grid to follow the

146motion, i.e. \f$\dot{r}=0\f$, so that the underbraced terms are

147dropped. This approach is generally known as a Lagrangian method, with

148the Lagrangian approach in MOM6 limited to the vertical

149direction. After each Lagrangian step, a regrid step is applied that

150generates a new vertical grid of the user's choosing. The ocean state

151is then remapped from the old to the new grid. The physical state is

152not meant to change during the remap step, yet truncation errors make

153remapping imperfect. We employ high-order accurate reconstructions to

154minimize errors introduced during the remap step (\cite white2008,

155\cite white2009). The connection between time-stepping and remapping

156is described in section \ref ALE_Timestep.

157

158*/