χ-ε relations

χ-ε relations#

Motivation#

Is \(K_T = K_ρ\) at 5-m scale a good assumption in spicy regions? We make this assumption to relate \(χ_T\) (\(K_T\)) to \(ε\) (\(K_ρ\))

We have some observations of \(K_T \sim O(1) K_S\) (Nash & Moum, 2002)
\(ρ = f(T, S)\) and T, S anomalies can compensate each other, i.e. you can have \(T\) anomalies that are passive and not connected to the energy budget, so one might suspect that it’s possible to get a larger \(χ_T\) for the same \(ε\) when T-S compensation is present vs absent

From buoyancy flux#

Define \(K =χ/2/x_z^2\) \(J=-\overline{w'x'} = - K x_z\)

Start with buoyancy flux \(J_b\) which we assume to be \(Γε\) later

(13)#\[\begin{align} J_ρ &= -\overline{w'ρ'} \\ &= -\overline{-w'ρ_0αT' + w'ρ_0βS'} \\ &= -ρ_0 (-α \overline{w'T'} + β \overline{w'S'}) \\ &= -ρ_0 (αJ_T - βJ_s) \\ &= -ρ_0 (-α K_T T_z + βK_S S_z) \end{align}\]

Signs seem right: sgn\((J_ρ) = -\)sgn\((J_T)\) and sign\((J_ρ)=\)sign\((J_S)\)

downward heat flux \(J_T < 0\) is upward density flux \(J_ρ > 0\) and upward \(J_s > 0\) is upward density flux \(J_ρ > 0\)

If \(K_T = K_S\) (e.g. Nash & Moum)

(14)#\[\begin{align} J_ρ &= ρ_0 (α K_T T_z - βK_S S_z) \\ &= -K_T ρ_0(-αT_z + βS_z) \\ &= -K_T ρ_z \end{align}\]

So \(K_T = K_ρ\) if \(K_ρ\) is defined so that \(J_ρ = -K_ρρ_z\)

To \(χ_ρ\)#

Gregg (1987)’s starting point is an expression for \(χ_{pe}\)

(15)#\[\begin{align} χ_{pe} &= \left(\frac{g}{ρN}\right)^2 χ_ρ \\ &= \left(\frac{gα}{N}\right)^2 χ_T + \left(\frac{gβ}{N}\right)^2 χ_S \qquad\qquad \text{assumption!} \end{align}\]

You can’t write \(χ_ρ\) using molecular gradients and diffusivity but we can make progress if we assume that \(χ_ρ = 2 K_ρ ρ_z^2\), analogous to \(χ_T\) Let’s try to derive this starting from \(J_ρ\).

(16)#\[\begin{align} χ_ρ &= -2J_ρ ρ_z \\ &= 2ρ_0 (-α K_T T_z + βK_S S_z) ρ_z \\ &= 2ρ_0 (-α K_T T_z + βK_S S_z) ρ_0 (-α T_z + β S_z) \\ &= 2ρ_0^2 (α^2 K_T T_z^2 + β^2 K_S S_z^2 - αβ K_T T_z S_z - αβ K_S S_z T_z) \\ &= ρ_0^2 \left[α^2 χ_T + β^2 χ_S - 2αβ (K_T + K_S) T_z S_z \right] \\ &= ρ_0^2 \left[α^2 χ_T + β^2 χ_S - \frac{4K_T α^2T_z^2}{R_ρ} \right] \\ &= ρ_0^2 \left[α^2 χ_T + β^2 χ_S - \frac{2α^2}{R_ρ} χ_T \right] \end{align}\]

So \(χ_{pe}\) is

(17)#\[\begin{align} χ_{pe} &= \left(\frac{g}{ρ_0N}\right)^2 χ_ρ \\ &= \left(\frac{gα}{N}\right)^2 χ_T + \left(\frac{gβ}{N}\right)^2 χ_S - \frac{2}{R_ρ}\left(\frac{gα}{N}\right)^2 χ_T \\ &= \left(\frac{gα}{N}\right)^2 χ_T \left( 1 -\frac{2}{R_ρ} + \frac{1}{R_ρ^2} \right) \\ &= \left(\frac{gα}{N}\right)^2 χ_T \left( \frac{R_ρ-1}{R_ρ}\right)^2 \end{align}\]

Then,

(18)#\[\begin{align} 2Γε &= \left(\frac{gα}{N}\right)^2 \left( \frac{R_ρ-1}{R_ρ}\right)^2 χ_T \\ \frac{Γε}{N^2} &= \left(\frac{gα}{N}\right)^2 \left( \frac{R_ρ-1}{R_ρ}\right)^2 \frac{χ_T/2}{T_z^2} \frac{T_z^2}{N^2}\\ K_ρ &= \left(\frac{gαT_z}{N^2}\right)^2 \left( \frac{R_ρ-1}{R_ρ}\right)^2 K_T \\ K_ρ &= \left(1 - \frac{1}{R_ρ}\right)^{-2} \left( \frac{R_ρ-1}{R_ρ}\right)^2 K_T \\ K_ρ &= K_T \end{align}\]

Well, so we know now that this is all consistent

Gregg (1987) & energetics#

An alternative way to formulate the \(χ\)-\(ε\) relation is to use energetics (connected to point 2) as in Gregg (1987); Seim & Gregg (1994); Alford & Pinkel (2000). Gergg (1987)’ equation (56) is the evolution equation for \(ρ'\)

(19)#\[\begin{equation} \frac{d\overline{ρ'^2}}{dt} + 2 \overline{ρ'u'} \frac{∂\overline{ρ}}{∂z} = - χ_ρ \end{equation}\]

Then form an equation for perturbation potential energy \(η\)

(20)#\[\begin{align} η &≡ \frac 12 \left( \frac{g}{ρN} \right)^2 \overline{ρ'^2} \\ \frac{dη}{dt} &= -J_b - \frac 12 χ_{pe} \end{align}\]

This shows the connection between \(ε\) (\(J_b\)) and \(χ_{pe}\) (tied to \(χ_ρ\) which should be related to \(χ_T\) and \(χ_S\)).

(21)#\[\begin{align} χ_{pe} &= \left(\frac{g}{ρN}\right)^2 χ_ρ \end{align}\]

What is \(χ_ρ\)#

Now \(χ_T\) and \(χ_S\) are “easy” to define (assuming isotropy):

(22)#\[\begin{equation} χ_T = 6 D_T \overline{\left(\frac{∂T'}{∂z}\right)^2} \qquad χ_S = 6 D_S \overline{\left(\frac{∂S'}{∂z}\right)^2} \end{equation}\]

where \(D_S, D_T\) are molecular diffusivities for \(S\), \(T\).

We cannot simply write \(χ_ρ = 6 \ D_ρ (∂ρ'/∂z)^2\): What is the “molecular diffusivity” of density? Instead \(χ_ρ\) must be the positive-definite dissipation term in the evolution equation of \(\overline{ρ'^2}\) derived from those for \(\overline{T'^2}\) and \(\overline{S'^2}\)

Approach of Gregg (1987)#

Gregg (1987) writes

(23)#\[\begin{align} χ_{pe} &= \left(\frac{g}{ρN}\right)^2 χ_ρ \\ &= \left(\frac{gα}{N}\right)^2 χ_T + \left(\frac{gβ}{N}\right)^2 χ_S \qquad\qquad \text{assumption!} \\ &= \left(\frac{gα}{N}\right)^2 \left( 1 + \frac{1}{R_ρ^2} \right) χ_T; \qquad R_ρ = \frac{αT_z}{βS_z} \; \text{ and } χ_S \approx \left( \frac{dS}{dT}\right)^2χ_T \end{align}\]

This must assume that \(T'\) and \(S'\) are uncorrelated. As Nash & Moum (1999) show, microconductivity observations (which is a different function of \(T, S\)) can have a very different spectral shape depending on the coherence between \(T'\), \(S'\) (see below)

Assuming equililbrium energetics \(χ_{pe} = 2Γε\) (Alford & Pinkel, 2000) yields a \(ε-χ_T\) relationship, that we can then write as a \(K_T -K_ρ\) relationship

(24)#\[\begin{align} 2Γε &= \left(\frac{gα}{N}\right)^2 \left( 1 + \frac{1}{R_ρ^2} \right) χ_T \\ \frac{Γε}{N^2} &= \left(\frac{gα}{N}\right)^2 \left( 1 + \frac{1}{R_ρ^2} \right) \frac{χ_T/2}{T_z^2} \frac{T_z^2}{N^2}\\ K_ρ &= \left(\frac{gαT_z}{N^2}\right)^2 \left( 1 + \frac{1}{R_ρ^2} \right) K_T \\ K_ρ &= \left(1 - \frac{1}{R_ρ}\right)^{-2} \left( 1 + \frac{1}{R_ρ^2} \right) K_T \\ K_ρ &= \frac{1 + R_ρ^2}{(1 - R_ρ)^2} K_T \qquad\text{ or } K_ρ = \frac{(βS_z)^2 + (αT_z)^2}{(βS_z - αT_z)^2} K_T \end{align}\]

after using

(25)#\[\begin{equation} \frac{N^2}{T_z} = gα \left(1 - \frac{βS_z}{αT_z} \right) = gα\left( 1 - \frac{1}{R_ρ} \right) \end{equation}\]

For \(ε\)

(26)#\[\begin{align} ε &= \frac{1 + R_ρ^2}{(1 - R_ρ)^2} \frac{N^2χ_T}{2ΓT_z^2} \\ &= \frac{1 + R_ρ^2}{(1 - R_ρ)^2} \left(1 - \frac{1}{R_ρ} \right) \frac{gαχ_T}{2ΓT_z} \\ &= \frac{1 + R_ρ^2}{(R_ρ - 1) R_ρ} \frac{gαχ_T}{2ΓT_z} \end{align}\]

Some properties:

This blows up at \(R_ρ=1 ⇒ αT_z = βS_z\) because then \(N^2 = gαT_z - gβS_z = 0\), so the gradients perfectly cancel each other. The \(ε\) equation is undefined (0/0) but I think it’s saying \(χ_T\) does not tell us anything about \(ε\) since \(T\) is totally disconnected from the energetics. This is the inverse of what I expected. But we do filter out small N2 anyway
When \(0 < R_ρ < 1\) , \(ε < 0\) but \(N^2 < 0\) so it gets filtered out.
If \(S_z = 0\), then \(K_T=K_ρ\)

What is \(R_ρ\) for NATRE? [right] Here’s the average of \(R_ρ\) along P surfaces across all profiles.

13 40
0  values == 0
28090

/Users/dcherian/mambaforge/envs/eddydiff/lib/python3.9/site-packages/numba/np/ufunc/gufunc.py:151: RuntimeWarning: invalid value encountered in get_gradient_sign
  return self.ufunc(*args, **kwargs)

[ ] put KT/Kρ for NATRE with curve evaluated for NATRE mean Rρ

It seems like this will reduce the magnitude of the ε bump by boosting vaues above/below the bump, and reducing the bump

(
    fxn_K(natre.Rρ.sel(pres=slice(250, None)))
    .where(np.abs(natre.Rρ) < 10)
    .coarsen(pres=10, boundary="trim")
    .mean()
    .mean(["latitude", "longitude"])
    .reset_coords(drop=True)
).hvplot(label="ε_new/ε_old", ylabel="").opts(logy=True, show_grid=True)

Deriving \(χ_ρ\)#

Really really messy!