HiPS Diffusivity

HiPS micromixing occurs between eddy events that change parcel states. This micromixing is implemented instantaneously, and parcel properties are brought to the mean of the involved parcels. This is the HiPS analog of molecular diffusion.

The usual model for diffusion is based on a random walk, in which a particle experiences a series of steps, each of length $\Delta$ and of frequency $1/\tau$. The diffusivity is then defined as $$D = \frac{1}{2}\frac{\Delta^2}{\tau}.$$ Note that $\Delta^2$ is the mean square displacement per step.

Micromixing occurs when eddy events happen at a level $i_s^*$ with length and time scales $l_s^*$ and $\tau_s^*$, respectively, which depend on the Schmidt number of the scalar.

Single scalar with $Sc\ge 1$

For a single scalar with $Sc\ge 1$, micromixing events occur between two neighboring parcels. The lengthscale of these parcels is $A^2l_s^*$, owing to eddy events at level $i_s^*$ swapping grandchild nodes or parcels. The micromixing to the mean consists of half the molecules constituting each parcel moving a distance $A^2l_s^*$ to the neighboring parcel, and half not moving.

The mean square displacement is then $\Delta^2=A^4l_s^{*2}/2$. Micromixing events happen at frequency $1/\tau_s^*$, with $\tau_s^*=\tau_*$ for $Sc\ge 1$. This gives $$D=\frac{A^4l_s^{*2}}{4\tau^*}.$$ For $A=1/2$ we have $$D=\frac{l_s^{*2}}{64\tau^*}.$$

Now, rewrite $D$ in terms of $L_0$, $\tau_0$, $N$, and $Sc$. Use

$$Sc = \left(\frac{l^*}{l_s^*}\right)^2,$$

$$l^* = L_0A^{N-3},$$ $$\tau^* = \tau_0\left(\frac{l^*}{L_0}\right)^{2/3}.$$ These give $$l_s^* = L_0A^{N-3}Sc^{-1/2},$$ $$\tau_* = \tau_0A^{\frac{2}{3}(N-3)},$$ which then give

$$\frac{D}{L_0^2/\tau_0} = \frac{A^{\frac{4N}{3}}}{4Sc}.$$

General Scalars

For scalars with $Sc\le 1$, the scalar *lives* at levels with smaller length scales (nearer the top of the tree), and micromixing mixes multiple parcels (powers of two) to their mean. See the figure below for a schematic. Similarly, scalars with $Sc\ge 1$ can have mixing across multiple parcels. This normally would happen if there are several scalars with $Sc\ge 1$ so that the scalar with the largest Sc nominally mixes between two parcels, but scalars with smaller $Sc$ (but still $Sc\ge 1$) may then mix between multiple parcels.

In the schematic shown, when micromixing to the mean of 8 parcels, 1/8 of the molecules in a given parcel move a distance 0, 1/8 move a distance $A^4l_s^*$, 2/8 move a distance $A^3l_s^*$, and 4/8 move a distance $A^4l_s^*$. The mean square distance is

$$\Delta^2 = 0^2\cdot\frac{1}{8} + (A^4l_s^*)^2\frac{1}{8} + (A^3l_s^*)^2\frac{2}{8} + (A^2l_s^*)^2\frac{4}{8}.$$

The frequency of these motions is $1/\tau_s^*$.

In general, $\Delta^2$ is given by

$$\Delta^2 = \frac{l_s^{*2}A^4}{2^{J+1}}\sum_{j=0}^J 2^j(A^2)^{J-j},$$

Where $J\equiv N_t-3-i_s^*$ is the number of levels between $i_s^*$ and the parcel grandparent node. Note, $N_t$ is the total number of levels in the tree, while $N$ is the number of levels needed to resolve a $Sc=1$ scalar. The above equation is simplified as follows:

$$\Delta^2 = \frac{l_s^{*2}A^4}{2\left(\frac{2}{A^2}\right)^J}\sum_{j=0}^J\left(\frac{2}{A^2}\right)^j,$$

$$\Delta^2 = \frac{l_s^{*2}A^4}{2\left(\frac{2}{A^2}\right)^J}\left(\frac{1-\left(\frac{2}{A^2}\right)^{J+1}}{1-\frac{2}{A^2}}\right),$$

$$\Delta^2 = \frac{l_s^{*2}A^4}{2}\left(\frac{\left(\frac{2}{A^2}\right)^{-J}-\frac{2}{A^2}}{1-\frac{2}{A^2}}\right).$$

While $i_s^*$ and $i^*$ are finite, the number of levels that can be considered in a tree may be infinite. That would accommodate arbitrary $Sc$ scalars. In the limit as $J\rightarrow\infty$, we have

$$\Delta^2 = \frac{l_s^{*2}A^4}{2-A^2}.$$

The diffusivity is given by $D=\Delta^2/2\tau_s^*$. We now write $D$ in terms of $L_0,$ $\tau_0,$ $A,$ $N,$ and $Sc$. Use

$$Sc=\left(\frac{l^*}{l_s^*}\right)^{p_s}\rightarrow l_s^* = l^*Sc^{-1/p_s} = L_0A^{N-3}Sc^{-1/p_s},$$

$$Sc\ge 1;\,\tau_s^*=\tau^* = \tau_0\left(\frac{l^*}{L_0}\right)^{2/3} = \tau_0A^{\frac{2}{3}(N-3)},$$ $$Sc\le 1;\,\tau_s^* = \tau_0\left(\frac{l_s^*}{L_0}\right)^{2/3} = \tau_0A^{\frac{2}{3}(N-3)}Sc^{-\frac{2}{3p_s}}.$$

These equations give

$$\frac{D}{L_0^2/\tau_0} = \frac{A^{\frac{4N}{3}}}{2(2-A^2)Sc}$$

for all $Sc$.

This diffusivity, on an infinite tree, can be compared to the diffusity on a minimal tree. The ratio of the green to blue equations is 8/7 $\approx$ 1.14 for $A=1/2$.

Inertial-diffusive range

In the inertial-diffusive range (where the scalar variance spectrum has slope -17/3), instead of mixing the respective two subtrees of the level-$i_s^*$ event apex, for any eddy event from levels $i_s^*$ to $i^*$, both the left and right subtrees of the event apex are individually mixed with probability $p$ based on a Bernoulli trial. But this is only effective if the subtrees are inhomogeneous, which occurs with probability $q^{j+1}=1-p$, where $j=j^\prime-i_s^*$ and $j^\prime$ indexes levels from $i_s^*$ to $i^*$ so that $j\in[0,i^*-i_s^*]$. The diffusivity is then given by

$$D = \sum_{j^\prime=i_s^*}^{i^*}D_{j^\prime},$$

$$D_{j^\prime} = \frac{pq^{j+1}\Delta_{j^\prime}^2}{2\tau_{j^\prime}}.$$

$\Delta_{j^\prime}^2$ is given, as above, by

$$\Delta_{j^\prime}^2 = \frac{l_{j^\prime}^2A^4}{2-A^2},$$

and $\tau_{j^\prime}$, and $l_{j^\prime}$ are

$$\tau_{j^\prime} = \tau_0\left(\frac{l_j}{L_0}\right)^{2/3},$$

$$l_{j^\prime} = L_0A^{j^\prime}.$$

These then give

$$\frac{D}{L_0^2/\tau_0} = \frac{pA^4A^{\frac{4}{3}i_s^*}}{2(2-A^2)}\sum_{j=0}^{i^*-i_s^*}(qA^{4/3})^j.$$

$$\frac{D}{L_0^2/\tau_0} = \frac{pA^4A^{\frac{4}{3}i_s^*}}{2(2-A^2)}\left(\frac{1-(qA^{4/3})^{i^*-i_s^*+1}}{1-qA^{4/3}}\right)$$

Now use $Sc=(l^*/l_s^*)^{4/3} = A^{\frac{4}{3}(i^*-i_s^*)}$, and $i^*=N-3$, to obtain

$$\frac{D}{L_0^2/\tau_0} = \frac{A^\frac{4N}{3}}{2(2-A^2)Sc}\cdot p\left(\frac{1-q^{i^*-i_s^*+1}A^{4/3}Sc}{1-qA^{4/3}}\right),$$

where $i^*-i_s^* = 3\log(Sc)/4\log(A)$.