39.7.1.1 Cabbeling, thermobaricity, and halobaricity

To isolate the mathematical expressions for cabbeling, thermobaricity, and halobaricity, write the convergence of the isoneutral diffusive fluxes in the form

$$-\rho_{\theta} \; \nabla \cdot \vec{F}_{I}(\theta) - \rho_{s} \; ... ... \rho_{\theta} \cdot \vec{F}_{I}(\theta) + \nabla \rho_{s} \cdot \vec{F}_{I}(s)$$ (39.57)

where the identity

 

$$\rho_{\theta} \; \vec{F}_{I}(\theta) + \rho_{s} \; \vec{F}_{I}(s) = 0$$ (39.58)

was used. This identity represents an important balance between the isoneutral diffusive flux of the two active tracers. It is a manifestation of the absence of a neutral direction diffusive flux of locally referenced potential density. More simply, isoneutral diffusion does not act on buoyancy, and equation (39.61) is a mathematical statement of this physical property. Griffies et al 1998 provide further discussion of this balance, and its importance when discretizing isoneutral diffusion. To procede, use the identities

  

$$\nabla s \cdot \vec{F}_{I}(\theta) \nabla \theta \cdot \vec{F}_{I}(s)$$ = (39.59)

$$\nabla \rho_{\theta} \rho_{\theta \theta} \nabla \theta + \rho_{\theta s}\nabla s + \rho_{\theta \, p} \nabla p$$ = (39.60)

$$\nabla \rho_{s} \rho_{s \theta} \nabla \theta + \rho_{s s}\nabla s + \rho_{s \, p} \nabla p.$$ = (39.61)

The first identity is most easy to verify when writing the diffusive flux in terms of a symmetric diffusion tensor

$$\nabla s \cdot \vec{F}_{I}(\theta) - \partial_{i}s \, K^{ij} \, \partial_{j}\theta$$ =

$$- \partial_{j}\theta \, K^{ji} \, \partial_{i}s$$ =

$$\nabla \theta \cdot \vec{F}_{I}(s).$$ = (39.62)

Identities (39.63) and (39.64) follow from the chain rule. The pressure gradient terms represent the effects of probing different pressure surfaces, and hence different potential density surfaces. Such is necessary for computing the spatial gradients of the thermal and saline expansion coefficients, and they lead to the thermobaric and halobaric effects. When evaluating the horizontal pressure gradient for this diagnostic, the gradient in the lid or surface pressure is ignored. These three identities, along with the balance of temperature and salinity fluxes given by equation (39.61), render

 

$${ -\rho_{\theta} \; \nabla \cdot \vec{F}_{I}(\theta) -\rho_{s} \; \nabla \cdot \vec{F}_{I}(s) =}$$

$$\rho_{\theta \theta} \nabla \theta \cdot \vec{F}_{I}(\theta) + \r... ...\cdot [ \rho_{\theta \, p} \vec{F}_{I}(\theta) + \rho_{s \, p} \vec{F}_{I}(s) ]$$

$$\nabla \theta \cdot \vec{F}_{I}(\theta) [ \rho_{\theta \theta} - ... ...cdot [ \rho_{\theta \, p} \vec{F}_{I}(\theta) + \rho_{s \, p} \vec{F}_{I}(s) ].$$ = (39.63)

The terms in equation (39.66) proportional to the second derivatives of density with respect to temperature and salinity are identified as cabbeling. There is a useful way to make the form for cabbeling appear in a slightly more tidy and intuitive manner. To do so, introduce the vector

$$\vec{V} (1,\alpha/\beta)$$ =

$$(1,-\rho_{\theta}/\rho_{s})$$ = (39.64)

and the symmetric tensor

$$\rho_{ab} = \left( \begin{array}{cc} \rho_{\theta \theta} & \rho_{\theta s} \\ \rho_{s \theta} & \rho_{s s} \end{array}\right),$$ (39.65)

which yields

 

$$\rho_{\theta \theta} \nabla \theta \cdot \vec{F}_{I}(\theta) + \r... ...a s \cdot \vec{F}_{I}(s) + 2 \rho_{\theta s} \nabla s \cdot \vec{F}_{I}(\theta) \nabla \theta \cdot \vec{F}_{I}(\theta) \, \rho_{a b} V^{a} V^{b}.$$ = (39.66)

The quadratic form

$$\rho_{a b} V^{a} V^{b} \rho_{\theta \theta} - 2 \rho_{s \theta} \; (\rho_{\theta}/ \rho_{s}) + \rho_{ss} \; (\rho_{\theta}/\rho_{s})^{2}$$ =

$$-\alpha_{\theta} - 2 (\alpha/\beta)\alpha_{s} + (\alpha /\beta)^{2}\beta_{s}$$ = (39.67)

can be thought of as the squared length of the vector $\vec{V}$ on the curved potential density surface characterized locally by the metric tensor $\rho_{a b}$. It can be easily computed in MOM through tabulating the first and second partial derivatives of the density (see Section 39.7.2.1). A fundamental property of seawater is that the total or Gaussian curvature (Aris, 1962) of a potential density surface

$$\kappa_{gauss} \det(\rho_{ab})(1+\rho_{\theta}^{2} + \rho_{s}^{2})^{-1}$$ =

$$(\rho_{\theta \theta} \, \rho_{ss} - \rho_{\theta s}^{2} ) (1+\rho_{\theta}^{2} + \rho_{s}^{2})^{-1}$$ = (39.68)

is negative. Consequently,

$$\rho_{a b} V^{a} V^{b} \le 0.$$ (39.69)

In summary, cabbeling

$$\mbox{cabbeling} = \nabla \theta \cdot \vec{F}_{I}(\theta) \; \, \rho_{a b} V^{a} V^{b}$$ (39.70)

is written as the product of two separate quadratic forms whose physical content can be individually identified. The first

$$\nabla \theta \cdot \vec{F}_{I}(\theta) = -\partial_{i} \theta \, K^{ij} \, \partial_{j}\theta$$ (39.71)

is the projection of the temperature gradient onto the isoneutral diffusive flux of temperature. This term is negative semi-definite for a downgradient isoneutral diffusive temperature flux. In this case, the tensor K^ij is the symmetric and positive semi-definite Redi isoneutral diffusion tensor. The second term, $\rho_{a b} V^{a} V^{b}$, as just discussed, summarizes certain intrinsic properties of the potential density surface. The combined effects of a negatively curved potential density surface and a downgradient isoneutral diffusive flux of temperature render

$$\mbox{cabbeling} \ge 0.$$ (39.72)

As a result, cabbeling increases the value of the locally referenced potential density $\rho$, thus moving a water parcel downward.

The terms proportional to the pressure gradient

$$\mbox{thermobaricity + halobaricity} \rho_{\theta \, p} \vec{F}_{I}(\theta) \cdot \nabla p + \rho_{s \, p} \vec{F}_{I}(s) \cdot \nabla p$$ =

$$\rho_{\theta \, p} \vec{F}_{I}(\theta) \cdot \nabla p - \rho_{s \... ...left( \frac{\rho_{\theta}}{\rho_{s}} \right) \vec{F}_{I}(\theta) \cdot \nabla p$$ = (39.73)

represent the effects from thermobaricity and halobaricity. The balance (39.61) was used to obtain the second equality. These processes arise from the pressure dependence of the equation of state for seawater. The ratio

$$\frac{ \mbox{thermobaricity} }{\mbox{halobaricity}} = - \left( \frac{ \rho_{\theta p} \, \rho_{s} }{ \rho_{s p} \, \rho_{\theta}} \right)$$ (39.74)

is quite large in absolute value, indicating the dominance of thermobaricity over halobaricity (McDougall 1987). In contrast to the cabbeling term, both the thermobaric and halobaric terms are sign-indefinite.