39.9.1 Thickness equation

For a stably stratified fluid, it is possible to introduce a general monotonic vertical coordinate

$$s = s(\lambda,\phi,z,t).$$ (39.120)

Conservation of volume (incompressible fluid) for a fluid parcel implies

$$\frac{D}{Dt} \, (a^{2} \, \cos\phi \, d\lambda \, d\phi \, dz) = \frac{D}{Dt} \, (a^{2} \, \cos\phi \, d\lambda \, d\phi \, z_{s} \, ds) = 0,$$ (39.121)

where D/Dt is the material time derivative. This relation leads to the thickness equation

 

$$\partial_{t} z_{s} + \nabla_{s} \cdot (\vec{u}_{h} \, z_{s}) + \partial_{s} (z_{s} \, \dot{s}) = 0,$$ (39.122)

where $\nabla_{s}$ is the lateral gradient operator taken with the generalized vertical coordinate s held fixed, and the dot represents a Lagrangian time derivative. $\vec{u}_{h}$ is the horizontal velocity vector

$$\vec{u}_{h} = (u,v) = (\dot{x},\dot{y}),$$ (39.123)

whereas $\dot{s}$ is the component of the velocity in the generalized vertical direction. In the direction perpendicular to the geopotential, the vertical velocity is given by

 

$$w = \dot{z} = (\partial_{t} + \vec{u}_{h} \cdot \nabla_{s} + \dot{s} \, \partial_{s} ) \, z.$$ (39.124)

In this expression, z_t is the time tendency for the depth of a constant s surface.

$$\nabla_{s}z = - \left( \frac{\nabla_{h} s}{s_{z}} \right)$$ (39.125)

is the horizontal slope vector of constant s surfaces, where $\nabla_{h}s$ is the horizontal gradient of s taken with depth zheld fixed, and s_z is the vertical gradient of the s surfaces. In general, if there is zero flow across the s surfaces, then $\dot{s} = 0$. Mathematically, z_s = (s_z)^-1 is the Jacobian of transformation between the (x,y,z,t) and the (x,y,s,t)coordinate systems. Often z_s is called the specific thickness. For a smooth and monotonic vertical coordinate, z_s is single signed and does not vanish. A trivial example for a vertical coordinate is s=z, which leads to z_s=1, $\dot{s} = \dot{z} = w$, and the thickness equation reduces to $\nabla \cdot \vec{u} = 0$, which is the continuity equation used in MOM. In the following, it will prove useful to write the thickness equation in one of the two following equivalent forms:

 

$$\partial_{s} (z_{t} + z_{s} \, \dot{s}) + \nabla_{s} \cdot (\vec{u}_{h} \, z_{s}) = 0,$$ (39.126)

or, upon insertion of the vertical velocity w from equation (39.127),

$$\partial_{s} (w - \vec{u}_{h} \cdot \nabla_{s}z) + \nabla_{s} \cdot (\vec{u}_{h} \, z_{s}) = 0.$$ (39.127)