Fundamentals for the energetics of Type II and Type III convection . 49

of N². Note that Sstrat, Scab and C_H^D_toH^P will only be non-zero if diabatic processes occur. Adiabatic processes, which define OCAPE, a↵ects Stb (see (2.21b)). However, Stb is also influenced by the diabatic processes since it is state-based.

In section 2.3 we derive the fundamentals of energy conservation and describe the numerical model. It would be very helpful for the reader to go through the funda- mentals of thermodynamics in section 2.3 to better capture the main points of this study. In section 2.4, we isolate and explain Stb and Sstrat using simplified simula- tions (excluding cabbeling in the EOS). In section 2.5 we increase the complexity of our simulation (using the full EOS) to evaluate and explain Scab and C_H^D_toH^P. We further propose a theory to predict the maximum depth of convection. In section 2.6, we apply this theory and (2.1) to a convection event initially based on a realistic profile from Weddell Sea. Section 2.7 comprises our discussion and conclusions. Nu- merical experiments (Tables 2.1-2.5; Figures 2.2, 2.4, 2.5, 2.6, 2.7) in this chapter is organized following step-by-step diagnosis for our energy decomposition, as stated by their titles.

C

conversion of H^D to H^P diabatically

H ^D H ^P

KE

dynamic part of PE

heat content part of PE

Energy transfer State change of H

^D

= H

^D

− H

^D

S _tb

thermobaricity (OCAPE)

cabbeling-induced volume reduction during mixing (descent of the center of mass )

mixing-induced stratification reduction (rise of the center of mass) initial state

ΔH !

^D

!!

_{final state}

S _cab ! -S _strat

source

H^DtoH^P!

Figure 2.1: Schematic of the proposed energetics for thermobaricity- and cabbeling- powered convection. Definitions and denotations here follow section 2.3.1. (left panel) Potential energy (PE) can be represented by the system’s enthalpyH, which includes the dynamic part H^D and the heat content part H^P (defined in (2.11a)–(2.11d)).

C_H^D_toH^P is the time-integrated energy transfer from theH^D reservoir to theH^P reser- voir diabatically (defined in (2.15b)). Heat_vis is the time-integrated viscous heating (defined in (2.15a)), which transfers energy from the KE reservoir to the H^P reser- voir. KEcum is the time-integrated work done by vertical buoyancy flux (defined in (2.14b)), which transfers energy from the H^D reservoir to the KE reservoir. Thus KEcum equals the current KE plus Heatvis, as well as equaling the state change of H^D minus C_H^D_toH^P (see (2.13b)). (right panel) The state change of H^D is due to three distinct sources/sinks: S_tb, S_cab, and S_strat (defined in (2.21b),(2.25),(2.20a), respectively). Therefore, KEcum has four contributions: C_H^D_toH^P, Stb, Scab, and Sstrat (i.e. (2.1)). The mathematical derivation of Equation (2.1) is provided in Appendix.

accelerations (see section 2.7.2 for associated discussion). By taking the curl of the momentum equation, we obtain the vorticity equation

Dr²

Dt = @b

@y +⌫r²r² , (2.2)

where⌫is the kinematic viscosity. More sophisticated schemes for turbulent viscosity could better parameterize subgrid turbulence. Here we adopt a Laplacian viscosity

because it is convenient for enforcing energy conservation. In equation (2.2) we use the modified buoyancy of Young (2010):

b=b(✓, S, P)⌘b(✓, S, z) = g(⇢ ⇢0)/⇢, (2.3)

where ✓, S, P and ⇢0 are potential temperature, salinity, pressure and constant ref- erence density, respectively. Here we replace P with z following the hydrostatic relation under Boussinesq approximation (Young, 2010). Following Equations (57.3) and (57.6) of Landau and Lifshitz (1959), we have the salinity and thermodynamic equations

DS

Dt = 1

⇢0r·i, (2.4a)

TD⌘

Dt = 1

⇢0

î r·(q µi) i·rµ+⇢0⌫(r² )^2ó, (2.4b)

where ⌘ is specific entropy, T is temperature, i is di↵usive salt flux, q is di↵usive energy flux,µis the relative chemical potential of salt in seawater, and ⇢0⌫(r² )² is viscous heating. Following ⌘ =⌘(✓, S) we rewrite (2.4b) as

TD⌘

Dt =Cp

D✓

Dt T µ✓

DS

Dt, (2.5a)

C_p = T@⌘

@✓ _S, µ_✓ = @µ

@T _S,Pr = @⌘

@S _✓, (2.5b)

following Maxwell’s relations. Here Pr is the reference pressure at sea level. Sub- stituting (2.4a) into (2.5a) and using (2.4b), we obtain the evolution equation for

✓

D✓

Dt = r·[q (µ T µ_✓)i]

Cp⇢0

i·r(µ T µ_✓) Cp⇢0

+⌫(r² )² Cp

. (2.6)

Note thatCp is proportional toT and is not a constant, as shown in (2.5b). Therefore viscous heating and di↵usion lead to the non-conservation of ✓, according to (2.6).

We demonstrate that equations (2.2), (2.3), (2.4a) and (2.6) (following Ingersoll, 2005; Young, 2010; Landau and Lifshitz, 1959) compose a non-hydrostatic energy- conserving (NHEC) model. From Part I, PE can be represented by the system’s en- thalpy (the energy in this chapter, if not otherwise stated, is always column-averaged and in units of J/kg):

PE =H = 1

RR ⇢0dydz

ZZ

h⇢0dydz , (2.7)

where h is specific enthalpy and has the following thermodynamic potential

@h

@✓ _S,P =Cp, @h

@S _✓,P = @h

@S _⌘,P + @h

@⌘ _S,P

@⌘

@S _✓ =µ T µ✓. (2.8)

where we have applied @h/@S|⌘,P = µ, @h/@⌘|S,P = T and (2.5b). One derives the energy conservation by multiplying (2.2) by ⇢₀ , multiplying (2.6) by ⇢₀@h/@✓ as expressed in (2.8), multiplying (2.4a) by ⇢0@h/@S as expressed in (2.8), and then

adding the result and integrating:

@

@t(KE +H) = 1

RR ⇢0dydz

ZZ ®

⇢₀ J( ,r² ) ⇢₀J( , h) r·q + ⇢0r·

ï

⌫r² r ⌫ rr²

ò´

dydz = 0, (2.9)

where J is the Jacobian. All terms on the right-hand side of (2.9) vanish provided there is no viscous stress, no normal velocity, and no di↵usion of energy at/across the boundaries. In deriving the second term on the right-hand side of (2.9) we have applied @h/@z|^S,✓ = b, J( , z) =w= @ /@y and integration by parts. Therefore the energy conservation of (2.9) is independent of the form of q, i, and the EOS of (2.3).

To close the NHEC model we follow equations (58.11) and (58.12) of Landau and Lifshitz (1959) and adopt the parameterization

i= ⇢0srS, q (µ T µ✓)i= ⇢0Cp0✓r✓, (2.10)

where Cp0 is a constant equal to 4000 J kg ^{1 o}C ¹ and s and ✓ are the kinematic di↵usivities of salt and heat, respectively. Equation (2.10) acts to parameterize the unresolved grid-scale turbulent di↵usion that tends to bring the fluid closer to an isohaline and isentropic state.

Only part of H (PE), called dynamic enthalpyH^D, contributes to the dynamics;

the remaining part ofH, called potential enthalpyH^P, represents the heat content of the system (Young, 2010; McDougall, 2003). In analogy to ✓, H^P is simply the sys-

tem’s enthalpy when all parcels are displaced adiabatically to the reference pressure.

H^P and H^D are defined as

H =H^P +H^D, H^P =

RR h^P⇢0dydz

RR ⇢0dydz , H^D =

RR h^D⇢0dydz

RR ⇢0dydz , (2.11a)

h^P(✓, S) = h(✓, S, P_r), (2.11b)

h^D(✓, S, P) =h(✓, S, P) h^P(✓, S) =

Z P

Pr

@h

@P⁰ _✓,SdP⁰ =

Z P

Pr

dP⁰

⇢(✓, S, P⁰) (2.11c)

= (P Pr)/⇢0+

Z ₀

z b(✓, S, z⁰)dz⁰. (2.11d)

AgainP is the hydrostatic pressure by using Boussinesq approximation (Young, 2010).

The domain integral of (P Pr)/⇢0 in (2.11d) is approximately constant and does not contribute to the evolution ofH^D (Young, 2010). The thermodynamic potentials of h^P and h^D are

C_p^P = @h^P

@✓ _S, C_p^D = @h^D

@✓ _S,z =

Z 0

z

@b

@✓ _S,z0dz⁰, (2.12a) µ^P = @h^P

@S _✓, µ^D = @h^D

@S _✓,z =

Z ₀

z

@b

@S _✓,z0dz⁰, (2.12b) C_p^P +C_p^D =Cp = T

✓C_p^P, µ^P +µ^D =µ T µ✓. (2.12c)

Equation (2.12c) follows from (2.5b) and (2.8), and uses T@⌘/@✓|S = (T /✓)C_p^P (Mc- Dougall, 2003).

By definition H^P can only be modified diabatically. By contrast, H^D relies on the vertical distribution of fluid and represents the gravitational PE (GPE), which is required to generate KE (see (2.14b) below). Similar to the derivation of (2.9), we

evaluate @h^D/@t and @h^P/@t in terms of @✓/@t and @S/@t (through thermodynamic potentials) and derive

@H^D

@t +@H^P

@t = @KE

@t , (2.13a)

@H^D

@t = 1

RR ⇢0dydz

ZZ @h^D

@t ⇢0dydz = @KE_cum

@t

@C_H^D_toH^P

@t , (2.13b)

@H^P

@t = 1

RR ⇢0dydz

ZZ @h^P

@t ⇢₀dydz = @Heatvis

@t + @C_H^D_toH^P

@t , (2.14a)

KEcum = KE + Heatvis = 1

RR ⇢₀dydz

Z t

0

ZZ

(wb)⇢0dydzdt, (2.14b)

Heatvis = 1

RR⇢0dydz

Z t 0

ZZ

⌫(r² )²⇢0dydzdt, (2.15a) CH^DtoH^P = 1

RR ⇢0dydz

Z t

0

ZZ "C_p^Dr·(Cp0✓r✓) Cp

+ C_p^D⌫(r² )² Cp

+ µ^Dr·(srS) + C_p^D_srS·r(µ^P +µ^D) Cp

#

⇢0dydzdt. (2.15b)

Heatvis is the cumulative viscous dissipation of KE (Figure 2.1). KEcum is the cu- mulative KE production by vertical buoyancy flux (derived from ⇥ (2.2)). C_H^D_toH^P is the time-integral of the rate of energy conversion of H^D to H^P (expressed using (2.10)), which depends on the unpredictable turbulent diabatic processes and can not be determined a priori (section 2.5.3).

In contrast to H^P, H^D contributes little to the system’s heat content, because

@h^D/@✓

@h^P/@✓ = C_p^D

C_p^P = T ✓

✓ <0.3%, (2.16)

following (2.12a) and (2.12c). Thus theH^D variation is insensitive to the nonconser- vation of ✓.

2.3.2 Numerical scheme

Equations (2.2), (2.3), (2.4a), (2.6), (2.10) define a closed system for numerical inte- gration. Throughout this chapter, except section 2.4, we use the full nonlinear EOS (Jackett et al., 2006). We computeh^P,h^D and their derivatives (µ^P,C_p^P,µ^D andC_p^D) using the state functions of Jackett et al. (2006). We use periodic boundaries inyand stress-free, zero-flux boundaries at the top/bottom. We discretize Laplacians using second-order-centered finite di↵erences. We compute Jacobians following Arakawa (1997). We use the Adams-Bashforth scheme (Press, 2007) for time integration. To resolve cabbeling instability, our default grid resolution is 0.83 m⇥0.83 m. To ensure numerical stability while minimally a↵ecting the turbulence, our default vertical and horizontal viscosity and tracer di↵usivity are 3⇥10 ⁴m²/s. This model conserves salinity to within the round-o↵error of the computer, and conserves energy to within 5% of the KE (PE+KE deviates by <5% of KE) in almost all simulations.

2.4 KE contributions from OCAPE and the reduction of stratification