plume_on_slope.rst

Gravity Plume On a Continental Slope

(in directory: verification/tutorial_plume_on_slope/)

Temperature after 23 hours of cooling. The cold dense water is mixed with ambient water as it accelerates down the slope and hence is warmer than the unmixed plume.

An important test of any ocean model is the ability to represent the flow of dense fluid down a slope. One example of such a flow is a non-rotating gravity plume on a continental slope, forced by a limited area of surface cooling above a continental shelf. Because the flow is non-rotating, a two dimensional model can be used in the across slope direction. The experiment is non-hydrostatic and uses open-boundaries to radiate transients at the deep water end. (Dense flow down a slope can also be forced by a dense inflow prescribed on the continental shelf; this configuration is being implemented by the DOME (Dynamics of Overflow Mixing and Entrainment) collaboration to compare solutions in different models).

The fluid is initially unstratified. The surface buoyancy loss B₀ (dimensions of L²T^− 3) over a cross-shelf distance R causes vertical convective mixing and modifies the density of the fluid by an amount

$$\Delta \rho = \frac{B_0 \rho_0 t}{g H}$$

where H is the depth of the shelf, g is the acceleration due to gravity, t is time since onset of cooling and ρ₀ is the reference density. Dense fluid slumps under gravity, with a flow speed close to the gravity wave speed:

$$U \sim \sqrt{g' H} \sim \sqrt{ \frac{g \Delta \rho H}{\rho_0} } \sim \sqrt{B_0 t}$$

A steady state is rapidly established in which the buoyancy flux out of the cooling region is balanced by the surface buoyancy loss. Then

$$U \sim (B_0 R)^{1/3} \mbox{ ; } \Delta \rho \sim \frac{\rho_0}{g H} (B_0 R)^{2/3}$$

The Froude number of the flow on the shelf is close to unity (but in practice slightly less than unity, giving subcritical flow). When the flow reaches the slope, it accelerates, so that it may become supercritical (provided the slope angle α is steep enough). In this case, a hydraulic control is established at the shelf break. On the slope, where the Froude number is greater than one, and gradient Richardson number (defined as Ri ∼ g′h^*/U² where h^* is the thickness of the interface between dense and ambient fluid) is reduced below 1/4, Kelvin-Helmholtz instability is possible, and leads to entrainment of ambient fluid into the plume, modifying the density, and hence the acceleration down the slope. Kelvin-Helmholtz instability is suppressed at low Reynolds and Peclet numbers given by

$$Re \sim \frac{U h}{ \nu} \sim \frac{(B_0 R)^{1/3} h}{\nu} \mbox{ ; } Pe = Re Pr$$

where h is the depth of the dense fluid on the slope. Hence this experiment is carried out in the high Re, Pe regime. A further constraint is that the convective heat flux must be much greater than the diffusive heat flux (Nusselt number  ≫ 1). Then

$$Nu = \frac{U h^* }{\kappa} \gg 1$$

Finally, since we have assumed that the convective mixing on the shelf occurs in a much shorter time than the horizontal equilibration, this implies H/R ≪ 1.

Hence to summarize the important non-dimensional parameters, and the limits we are considering:

$$\frac{H}{R} \ll 1 \mbox{ ; } Re \gg 1 \mbox{ ; } Pe \gg 1 \mbox{ ; } Nu \gg 1 \mbox{ ; } \mbox{ ; } Ri < 1/4 .$$

In addition we are assuming that the slope is steep enough to provide sufficient acceleration to the gravity plume, but nonetheless much less that 1:1, since many Kelvin-Helmholtz billows appear on the slope, implying horizontal length scale of the slope ≫ the depth of the dense fluid.

Configuration

The topography, spatial grid, forcing and initial conditions are all specified in binary data files generated using matlab script verification/tutorial_plume_on_slope/input/gendata.m and detailed in tut_plume_on_slope_config. Other model parameters are specified in input/data <verification/tutorial_plume_on_slope/input/data> and input/data.obcs <verification/tutorial_plume_on_slope/input/data.obcs> and detailed in tut_plume_on_slope_mod_parms.

Binary input data

Horizontal grid spacing, Δx, in the across-slope direction for the gravity plume experiment.

Topography, h(x), used for the gravity plume experiment.

Upward surface heat flux, Q(x), used as forcing in the gravity plume experiment.

The domain is 200 m deep and 6.4 km across. Uniform resolution of 60 × 3¹/₃ m is used in the vertical and variable resolution of the form shown in dx-plume-on-slope with 320 points is used in the horizontal. The formula for Δx is:

$$\Delta x(i) = \Delta x_1 + ( \Delta x_2 - \Delta x_1 ) ( 1 + \tanh{\left(\frac{i-i_s}{w}\right)} ) /2$$

where

$$\begin{aligned} \begin{aligned} Nx & = 320 \\\ Lx & = 6400 \;\; \mbox{(m)} \\\ \Delta x_1 & = \frac{2}{3} \frac{Lx}{Nx} \;\; \mbox{(m)} \\\ \Delta x_2 & = \frac{Lx/2}{Nx-Lx/(2 \Delta x_1)} \;\; \mbox{(m)} \\\ i_s & = Lx/( 2 \Delta x_1 ) \\\ w & = 40\end{aligned} \end{aligned}$$

Here, Δx₁ is the resolution on the shelf, Δx₂ is the resolution in deep water and Nx is the number of points in the horizontal.

The topography, shown in depth-plume-on-slope, is given by:

$$H(x) = -H_o + (H_o - h_s) ( 1 + \tanh{\left(\frac{x-x_s}{L_s}\right)} ) / 2$$

where

$$\begin{aligned} \begin{aligned} H_o & = 200 \;\; \mbox{(m)} \\\ h_s & = 40 \;\; \mbox{(m)} \\\ x_s & = 1500 + Lx/2 \;\; \mbox{(m)} \\\ L_s & = \frac{(H_o - h_s)}{2 s} \;\; \mbox{(m)} \\\ s & = 0.15\end{aligned} \end{aligned}$$

Here, s is the maximum slope, H_o is the maximum depth, h_s is the shelf depth, x_s is the lateral position of the shelf-break and L_s is the length-scale of the slope.

The forcing is through heat loss over the shelf, shown in Q-plume-on-slope and takes the form of a fixed flux with profile:

$$Q(x) = Q_o ( 1 + \tanh{\left(\frac{x - x_q}{L_q}\right)} ) / 2$$

where

$$\begin{aligned} \begin{aligned} Q_o & = 200 \;\; \mbox{(W m$^{-2}$)} \\\ x_q & = 2500 + Lx/2 \;\; \mbox{(m)} \\\ L_q & = 100 \;\; \mbox{(m)}\end{aligned} \end{aligned}$$

Here, Q_o is the maximum heat flux, x_q is the position of the cut-off, and L_q is the width of the cut-off.

The initial temperature field is unstratified but with random perturbations, to induce convection early on in the run. The random perturbation are calculated in computational space and because of the variable resolution introduce some spatial correlations, but this does not matter for this experiment. The perturbations have range 0 − 0.01 ^∘K.

Code configuration

The computational domain (number of gridpoints) is specified in code/SIZE.h <verification/tutorial_plume_on_slope/code/SIZE.h> and is configured as a single tile of dimensions 320 × 1 × 60.

To compile the model code for this experiment, the non-hydrostatic algorithm needs to be enabled, and the open-boundaries package (pkg/obcs) is required:

• Non-hydrostatic terms and algorithm are enabled with #define ALLOW_NONHYDROSTATIC in code/CPP_OPTIONS.h <verification/tutorial_plume_on_slope/code/CPP_OPTIONS.h> and activated with nonHydrostatic =.TRUE, in namelist PARM01 of input/data <verification/tutorial_plume_on_slope/input/data>.
• Open boundaries are enabled by adding line obcs to package configuration file code/packages.conf <verification/tutorial_plume_on_slope/code/packages.conf> and activated via useOBCS =.TRUE, in namelist PACKAGES of input/data.pkg <verification/tutorial_plume_on_slope/input/data.pkg>.

Model parameters

Model parameters used in the gravity plume experiment.

Parameter	Value	Description
g	9.81 m s-2	acceleration due to gravity
ρo	999.8 kg m-3	reference density
α	2 × 10-4 K-1	expansion coefficient
Ah	1 × 10-2 m2 s-1	horizontal viscosity
Av	1 × 10-3 m2 s-1	vertical viscosity
κh	0 m2 s-1	(explicit) horizontal diffusion
κv	0 m2 s-1	(explicit) vertical diffusion
Δt	20 s	time step
Δz	3.33333 m	vertical grid spacing
Δx	13.3333 - 39.5 m	horizontal grid spacing

The model parameters (tut_plume_parm_table) are specified in input/data <verification/tutorial_plume_on_slope/input/data> and if not assume the default values as defined in customize_model. A linear equation of state is used, eosType =’LINEAR’, but only temperature is active, sBeta =0.E-11. For the given heat flux, Q_o, the buoyancy forcing is $B_o = \frac{g \alpha Q}{\rho_o c_p} \sim 10^{-7}$ m² s^-3. Using R = 10³ m, the shelf width, this gives a velocity scale of U ∼ 5 × 10^− 2 m s^-1 for the initial front but will accelerate by an order of magnitude over the slope. The temperature anomaly will be of order Δθ ∼ 3 × 10^− 2 K. The viscosity is constant and gives a Reynolds number of 100, using h = 20 m for the initial front and will be an order magnitude bigger over the slope. There is no explicit diffusion but a non-linear advection scheme is used for temperature which adds enough diffusion so as to keep the model stable. The time-step is set to 20 s and gives Courant number order one when the flow reaches the bottom of the slope.