# Single-Mode Solutions for Convection and Double-Diffusive Convection in Porous Media

^{*}

## Abstract

**:**

## 1. Introduction

_{2}) in large porous underground reservoirs to mitigate CO

_{2}emissions [36,37].

## 2. Single-Mode Solutions for Double-Diffusive Convection in a Porous Medium

## 3. Comparisons of Single-Mode Solutions with Direct Numerical Simulations

#### 3.1. Thermal Convection with Passive Salinity ${R}_{\rho}=0$

#### 3.2. Double-Diffusive Convection with ${R}_{\rho}\ne 0$

## 4. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

DNS | Direct numerical simulations |

RBC | Rayleigh-Bénard convection |

ODDC | Oscillatory double-diffusive convection |

B.C. | Boundary conditions |

NBC | Neumann boundary conditions |

CO_{2} | Carbon dioxide |

2D | Two-dimensional |

3D | Three-dimensional |

c.c. | complex conjugate |

RMS | Root mean square |

NLBVP | Nonlinear boundary value problem |

SW | Standing waves |

TW | Traveling waves |

## References

- Herring, J.R. Investigation of problems in thermal convection. J. Atmos. Sci.
**1963**, 20, 325–338. [Google Scholar] [CrossRef] - Herring, J.R. Investigation of problems in thermal convection: Rigid boundaries. J. Atmos. Sci.
**1964**, 21, 277–290. [Google Scholar] [CrossRef] - Elder, J.W. The temporal development of a model of high Rayleigh number convection. J. Fluid Mech.
**1969**, 35, 417–437. [Google Scholar] [CrossRef] - Gough, D.O.; Spiegel, E.A.; Toomre, J. Modal equations for cellular convection. J. Fluid Mech.
**1975**, 68, 695–719. [Google Scholar] [CrossRef] - Toomre, J.; Gough, D.O.; Spiegel, E.A. Numerical solutions of single-mode convection equations. J. Fluid Mech.
**1977**, 79, 1–31. [Google Scholar] [CrossRef] - Gough, D.O.; Toomre, J. Single-mode theory of diffusive layers in thermohaline convection. J. Fluid Mech.
**1982**, 125, 75–97. [Google Scholar] [CrossRef] - Turner, J.S. The coupled turbulent transports of salt and and heat across a sharp density interface. Int. J. Heat Mass Transf.
**1965**, 8, 759–767. [Google Scholar] [CrossRef] - Crapper, P.F. Measurements across a diffusive interface. Deep Sea Res. Oceanogr. Abstr.
**1975**, 22, 537–545. [Google Scholar] [CrossRef] - Marmorino, G.O.; Caldwell, D.R. Heat and salt transport through a diffusive thermohaline interface. Deep Sea Res. Oceanogr. Abstr.
**1976**, 23, 59–67. [Google Scholar] [CrossRef] - Paparella, F.; Spiegel, E.A.; Talon, S. Shear and mixing in oscillatory doubly diffusive convection. Geophys. Astrophys. Fluid Dyn.
**2002**, 96, 271–289. [Google Scholar] [CrossRef] - Paparella, F.; Spiegel, E.A. Sheared salt fingers: Instability in a truncated system. Phys. Fluids
**1999**, 11, 1161–1168. [Google Scholar] [CrossRef] - Liu, C.; Julien, K.; Knobloch, E. Staircase solutions and stability in vertically confined salt-finger convection. J. Fluid Mech.
**2022**, 952, A4. [Google Scholar] [CrossRef] - Blennerhassett, P.J.; Bassom, A.P. Nonlinear high-wavenumber Bénard convection. IMA J. Appl. Math.
**1994**, 52, 51–77. [Google Scholar] [CrossRef] - Lewis, S.; Rees, D.A.S.; Bassom, A.P. High wavenumber convection in tall porous containers heated from below. Q. J. Mech. Appl. Math.
**1997**, 50, 545–563. [Google Scholar] [CrossRef] - Proctor, M.R.E.; Holyer, J.Y. Planform selection in salt fingers. J. Fluid Mech.
**1986**, 168, 241–253. [Google Scholar] [CrossRef] - Julien, K.; Knobloch, E. Reduced models for fluid flows with strong constraints. J. Math. Phys.
**2007**, 48, 065405. [Google Scholar] [CrossRef] [Green Version] - Taylor, G.I. Motion of solids in fluids when the flow is not irrotational. Proc. R. Soc. Lond. Ser. A
**1917**, 93, 99–113. [Google Scholar] - Proudman, J. On the motion of solids in a liquid possessing vorticity. Proc. R. Soc. Lond. Ser. A
**1916**, 92, 408–424. [Google Scholar] - Sprague, M.; Julien, K.; Knobloch, E.; Werne, J. Numerical simulation of an asymptotically reduced system for rotationally constrained convection. J. Fluid Mech.
**2006**, 551, 141–174. [Google Scholar] [CrossRef] - Grooms, I.; Julien, K.; Weiss, J.B.; Knobloch, E. Model of convective Taylor columns in rotating Rayleigh-Bénard convection. Phys. Rev. Lett.
**2010**, 104, 224501. [Google Scholar] [CrossRef] - Calkins, M.A.; Julien, K.; Tobias, S.M.; Aurnou, J.M.; Marti, P. Convection-driven kinematic dynamos at low Rossby and magnetic Prandtl numbers: Single mode solutions. Phys. Rev. E
**2016**, 93, 023115. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Calkins, M.A.; Long, L.; Nieves, D.; Julien, K.; Tobias, S.M. Convection-driven kinematic dynamos at low Rossby and magnetic Prandtl numbers. Phys. Rev. Fluids
**2016**, 1, 083701. [Google Scholar] [CrossRef] - Yang, Y.; van der Poel, E.P.; Ostilla-Mónico, R.; Sun, C.; Verzicco, R.; Grossmann, S.; Lohse, D. Salinity transfer in bounded double diffusive convection. J. Fluid Mech.
**2015**, 768, 476–491. [Google Scholar] [CrossRef] [Green Version] - Xie, J.H.; Julien, K.; Knobloch, E. Jet formation in salt-finger convection: A modified Rayleigh–Bénard problem. J. Fluid Mech.
**2019**, 858, 228–263. [Google Scholar] [CrossRef] - Hewitt, D.R.; Neufeld, J.A.; Lister, J.R. Ultimate regime of high Rayleigh number convection in a porous medium. Phys. Rev. Lett.
**2012**, 108, 224503. [Google Scholar] [CrossRef] [Green Version] - Wen, B.; Corson, L.T.; Chini, G.P. Structure and stability of steady porous medium convection at large Rayleigh number. J. Fluid Mech.
**2015**, 772, 197–224. [Google Scholar] [CrossRef] [Green Version] - Hewitt, D.R.; Neufeld, J.A.; Lister, J.R. High Rayleigh number convection in a three-dimensional porous medium. J. Fluid Mech.
**2014**, 748, 879–895. [Google Scholar] [CrossRef] [Green Version] - Pirozzoli, S.; De Paoli, M.; Zonta, F.; Soldati, A. Towards the ultimate regime in Rayleigh–Darcy convection. J. Fluid Mech.
**2021**, 911, R4. [Google Scholar] [CrossRef] - Hewitt, D.R. Vigorous convection in porous media. Proc. R. Soc. A
**2020**, 476, 20200111. [Google Scholar] [CrossRef] - Nield, D.A.; Bejan, A. Convection in Porous Media; Springer: Berlin/Heidelberg, Germany, 2006. [Google Scholar]
- Vafai, K. Handbook of Porous Media; CRC Press: Boca Raton, FL, USA, 2015. [Google Scholar]
- Mojtabi, A.; Charrier-Mojtabi, M.C. Double-diffusive convection in porous media. In Handbook of Porous Media; CRC Press: Boca Raton, FL, USA, 2005; pp. 287–338. [Google Scholar]
- Cheng, P. Heat transfer in geothermal systems. In Advances in Heat Transfer; Elsevier: Amsterdam, The Netherlands, 1979; Volume 14, pp. 1–105. [Google Scholar]
- Wooding, R.A.; Tyler, S.W.; White, I. Convection in groundwater below an evaporating salt lake: 1. Onset of instability. Water Resour. Res.
**1997**, 33, 1199–1217. [Google Scholar] [CrossRef] - Wooding, R.A.; Tyler, S.W.; White, I.; Anderson, P.A. Convection in groundwater below an evaporating salt lake: 2. Evolution of fingers or plumes. Water Resour. Res.
**1997**, 33, 1219–1228. [Google Scholar] [CrossRef] - Huppert, H.E.; Neufeld, J.A. The fluid mechanics of carbon dioxide sequestration. Annu. Rev. Fluid Mech.
**2014**, 46, 255–272. [Google Scholar] [CrossRef] - Neufeld, J.A.; Hesse, M.A.; Riaz, A.; Hallworth, M.A.; Tchelepi, H.A.; Huppert, H.E. Convective dissolution of carbon dioxide in saline aquifers. Geophys. Res. Lett.
**2010**, 37, L22404. [Google Scholar] [CrossRef] - Caltagirone, J.P. Thermoconvective instabilities in a horizontal porous layer. J. Fluid Mech.
**1975**, 72, 269–287. [Google Scholar] [CrossRef] - Trevisan, O.V.; Bejan, A. Mass and heat transfer by high Rayleigh number convection in a porous medium heated from below. Int. J. Heat Mass Transf.
**1987**, 30, 2341–2356. [Google Scholar] [CrossRef] - Rosenberg, N.D.; Spera, F.J. Thermohaline convection in a porous medium heated from below. Int. J. Heat Mass Transf.
**1992**, 35, 1261–1273. [Google Scholar] [CrossRef] - Goyeau, B.; Songbe, J.P.; Gobin, D. Numerical study of double-diffusive natural convection in a porous cavity using the Darcy-Brinkman formulation. Int. J. Heat Mass Transf.
**1996**, 39, 1363–1378. [Google Scholar] [CrossRef] - Mamou, M.; Vasseur, P.; Bilgen, E. Double-diffusive convection instability in a vertical porous enclosure. J. Fluid Mech.
**1998**, 368, 263–289. [Google Scholar] [CrossRef] - Mamou, M.; Vasseur, P. Thermosolutal bifurcation phenomena in porous enclosures subject to vertical temperature and concentration gradients. J. Fluid Mech.
**1999**, 395, 61–87. [Google Scholar] [CrossRef] - Mahidjiba, A.; Mamou, M.; Vasseur, P. Onset of double-diffusive convection in a rectangular porous cavity subject to mixed boundary conditions. Int. J. Heat Mass Transf.
**2000**, 43, 1505–1522. [Google Scholar] [CrossRef] - Bahloul, A.; Boutana, N.; Vasseur, P. Double-diffusive and Soret-induced convection in a shallow horizontal porous layer. J. Fluid Mech.
**2003**, 491, 325–352. [Google Scholar] [CrossRef] - Knobloch, E.; Deane, A.E.; Toomre, J.; Moore, D.R. Doubly diffusive waves. Contemp. Math.
**1986**, 56, 203–216. [Google Scholar] - Deane, A.E.; Knobloch, E.; Toomre, J. Traveling waves and chaos in thermosolutal convection. Phys. Rev. A
**1987**, 36, 2862–2869. [Google Scholar] [CrossRef] [PubMed] - Knobloch, E. Oscillatory convection in binary mixtures. Phys. Rev. A
**1986**, 34, 1538–1549. [Google Scholar] [CrossRef] - Knobloch, E.; Moore, D.R. Minimal model of binary fluid convection. Phys. Rev. A
**1990**, 42, 4693–4709. [Google Scholar] [CrossRef] - Predtechensky, A.A.; McCormick, W.D.; Swift, J.B.; Rossberg, A.G.; Swinney, H.L. Traveling wave instability in sustained double-diffusive convection. Phys. Fluids
**1994**, 6, 3923–3935. [Google Scholar] [CrossRef] [Green Version] - Predtechensky, A.A.; McCormick, W.D.; Swift, J.B.; Noszticzius, Z.; Swinney, H.L. Onset of traveling waves in isothermal double diffusive convection. Phys. Rev. Lett.
**1994**, 72, 218–221. [Google Scholar] [CrossRef] - Otero, J.; Dontcheva, L.A.; Johnston, H.; Worthing, R.A.; Kurganov, A.; Petrova, G.; Doering, C.R. High-Rayleigh-number convection in a fluid-saturated porous layer. J. Fluid Mech.
**2004**, 500, 263–281. [Google Scholar] [CrossRef] [Green Version] - Stern, M.E. Collective instability of salt fingers. J. Fluid Mech.
**1969**, 35, 209–218. [Google Scholar] [CrossRef] - Holyer, J.Y. The stability of long, steady, two-dimensional salt fingers. J. Fluid Mech.
**1984**, 147, 169–185. [Google Scholar] [CrossRef] - Radko, T. Double-Diffusive Convection; Cambridge University Press: Cambridge, MA, USA, 2013. [Google Scholar]
- Uecker, H.; Wetzel, D.; Rademacher, J.D.M. pde2path-A Matlab package for continuation and bifurcation in 2D elliptic systems. Numer. Math. Theory Methods Appl.
**2014**, 7, 58–106. [Google Scholar] [CrossRef] - Uecker, H. Numerical Continuation and Bifurcation in Nonlinear PDEs; Springer: Berlin/Heidelberg, Germany, 2021. [Google Scholar]
- Weideman, J.A.; Reddy, S.C. A MATLAB differentiation matrix suite. ACM Trans. Math. Software
**2000**, 26, 465–519. [Google Scholar] [CrossRef] [Green Version] - Uecker, H. pde2path without Finite Elements. 2021. Available online: http://www.staff.uni-oldenburg.de/hannes.uecker/pde2path/tuts/modtut.pdf (accessed on 12 December 2021).
- Burns, K.J.; Vasil, G.M.; Oishi, J.S.; Lecoanet, D.; Brown, B.P. Dedalus: A flexible framework for numerical simulations with spectral methods. Phys. Rev. Res.
**2020**, 2, 023068. [Google Scholar] [CrossRef] - Bassom, A.P.; Zhang, K. Strongly nonlinear convection cells in a rapidly rotating fluid layer. Geophys. Astrophys. Fluid Dyn.
**1994**, 76, 223–238. [Google Scholar] [CrossRef] - Rademacher, J.D.M.; Uecker, H. Symmetries, Freezing, and Hopf Bifurcations of Traveling Waves in pde2path. 2017. Available online: https://www.staff.uni-oldenburg.de/hannes.uecker/pde2path/tuts/symtut.pdf (accessed on 15 December 2021).
- Turner, J.S. Buoyancy Effects in Fluids; Cambridge University Press: Cambridge, UK, 1979. [Google Scholar]
- Lecoanet, D.; Le Bars, M.; Burns, K.J.; Vasil, G.M.; Brown, B.P.; Quataert, E.; Oishi, J.S. Numerical simulations of internal wave generation by convection in water. Phys. Rev. E
**2015**, 91, 063016. [Google Scholar] [CrossRef] [Green Version] - Lecoanet, D.; Quataert, E. Internal gravity wave excitation by turbulent convection. Mon. Not. R. Astron. Soc.
**2013**, 430, 2363–2376. [Google Scholar] [CrossRef] [Green Version] - Lecoanet, D.; Schwab, J.; Quataert, E.; Bildsten, L.; Timmes, F.X.; Burns, K.J.; Vasil, G.M.; Oishi, J.S.; Brown, B.P. Turbulent chemical diffusion in convectively bounded carbon flames. Astrophys. J.
**2016**, 832, 71. [Google Scholar] [CrossRef] - Couston, L.A.; Lecoanet, D.; Favier, B.; Le Bars, M. Dynamics of mixed convective–stably-stratified fluids. Phys. Rev. Fluids
**2017**, 2, 094804. [Google Scholar] [CrossRef] - Le Bars, M.; Lecoanet, D.; Perrard, S.; Ribeiro, A.; Rodet, L.; Aurnou, J.M.; Le Gal, P. Experimental study of internal wave generation by convection in water. Fluid Dyn. Res.
**2015**, 47, 045502. [Google Scholar] [CrossRef] [Green Version] - Couston, L.A.; Lecoanet, D.; Favier, B.; Le Bars, M. The energy flux spectrum of internal waves generated by turbulent convection. J. Fluid Mech.
**2018**, 854, R3. [Google Scholar] [CrossRef] [Green Version] - Bouffard, M.; Favier, B.; Lecoanet, D.; Le Bars, M. Internal gravity waves in a stratified layer atop a convecting liquid core in a non-rotating spherical shell. Geophys. J. Int.
**2022**, 228, 337–354. [Google Scholar] [CrossRef] - Léard, P.; Favier, B.; Le Gal, P.; Le Bars, M. Coupled convection and internal gravity waves excited in water around its density maximum at 4° C. Phys. Rev. Fluids
**2020**, 5, 024801. [Google Scholar] [CrossRef] - Le Bars, M.; Couston, L.A.; Favier, B.; Léard, P.; Lecoanet, D.; Le Gal, P. Fluid dynamics of a mixed convective/stably stratified system—A review of some recent works. Comptes Rendus. Phys.
**2020**, 21, 151–164. [Google Scholar] [CrossRef] - Aidun, C.K.; Steen, P.H. Transition to oscillatory convective heat transfer in a fluid-saturated porous medium. J. Thermophys. Heat Transf.
**1987**, 1, 268–273. [Google Scholar] [CrossRef] - Graham, M.D.; Steen, P.H. Plume formation and resonant bifurcations in porous-media convection. J. Fluid Mech.
**1994**, 272, 67–90. [Google Scholar] [CrossRef] - Graham, M.D.; Steen, P.H. Strongly interacting travelling waves and quasiperiodic dynamics in porous medium convection. Phys. D
**1992**, 54, 331–350. [Google Scholar] [CrossRef] - Julien, K.; Knobloch, E. Fully nonlinear oscillatory convection in a rotating layer. Phys. Fluids
**1997**, 9, 1906–1913. [Google Scholar] [CrossRef] [Green Version] - Knobloch, E.; Silber, M. Travelling wave convection in a rotating layer. Geophys. Astrophys. Fluid Dyn.
**1990**, 51, 195–209. [Google Scholar] [CrossRef] - Knobloch, E.; Proctor, M.R.E. Nonlinear periodic convection in double-diffusive systems. J. Fluid Mech.
**1981**, 108, 291–316. [Google Scholar] [CrossRef] - Dangelmayr, G.; Knobloch, E. The Takens-Bogdanov bifurcation with O(2)-symmetry. Philos. Trans. R. Soc. Lond. Ser. A
**1987**, 322, 243–279. [Google Scholar] - Greene, J.M.; Kim, J.S. The steady states of the Kuramoto-Sivashinsky equation. Phys. D
**1988**, 33, 99–120. [Google Scholar] [CrossRef] - Batiste, O.; Knobloch, E.; Alonso, A.; Mercader, I. Spatially localized binary-fluid convection. J. Fluid Mech.
**2006**, 560, 149–158. [Google Scholar] [CrossRef]

**Figure 1.**RMS profiles of (

**a**) ${T}_{\mathrm{rms}}\left(z\right)$, (

**b**) ${w}_{\mathrm{rms}}\left(z\right)$ and (

**c**) ${u}_{\mathrm{rms}}\left(z\right)$ at $R{a}_{T}=4000$, 8000, 16,000 associated with the 3D wavenumber ${k}_{x}={k}_{y}=0.17R{a}_{T}^{0.52}$ obtained from single-mode solutions (lines) compared with DNS results (Figure 8b in [27]) (lines with markers). Legend for all three panels is provided in panel (

**b**).

**Figure 2.**Comparisons of the mean temperature profiles obtained from single-mode solutions (lines) with DNS (lines with markers). Panel (

**a**) displays 2D results at $R{a}_{T}=10,000$, 20,000 and 40,000 using ${k}_{x}=0.48R{a}_{T}^{0.4}$ and ${k}_{y}=0$ compared with 2D DNS (Figure 3a in [25]). Panel (

**b**) shows 3D results at $R{a}_{T}=4000$, 8000 and 16,000 using ${k}_{x}={k}_{y}=0.17R{a}_{T}^{0.52}$ compared with 3D DNS (Figure 7 in [27]). Panels (

**c**) and (

**d**) show zooms of panels (

**a**) and (

**b**) near the bottom boundary, respectively.

**Figure 4.**$Nu$ as a function of $R{a}_{T}$ from single-mode solutions (black lines). Panel (

**a**) shows 2D results with ${k}_{x}=0.48R{a}_{T}^{0.4}$, ${k}_{y}=0$ compared with DNS data (Figure 2 in [25] and Figure 5b in [26]), exact 2D steady convection rolls (Figure 5b in [26]), and upper bound theory (Figure 5 in [52]). Panel (

**b**) displays 3D results with ${k}_{x}={k}_{y}=0.17R{a}_{T}^{0.52}$ compared with DNS data (Figure 2a in [27] and Table 1 [28]) as well as upper bound theory (Figure 5 in [52]). The single-mode solutions are stable within this severe truncation.

**Figure 5.**$Nu$ as a function of the horizontal wavenumber ${k}_{x}$ for $R{a}_{T}=50$, 100, 200, 400, 1000 and 2000 (from bottom to top) obtained from single-mode solutions (lines) and compared with DNS results (black squares) with ${k}_{x}=n\pi /{L}_{x,\mathrm{NBC}}$ using n and ${L}_{x,\mathrm{NBC}}$ appropriate to no-flux horizontal B.C. (Figure 4 and Table 2 in [39]). Panel (

**b**) is a zoom of panel (

**a**).

**Figure 6.**Solution profile of single-mode solutions displaying isocontours of (

**a**) streamfunction $\psi $, (

**b**) total temperature $1-z+T$ and (

**c**) total salinity $1-z+S$ at $R{a}_{T}=200$, ${k}_{x}=1.89\pi $, and $Le=4$. Panel (

**d**) shows the isocontours of total salinity $1-z+S$ at $Le=20$ with other parameters unchanged. This figure is to be compared with the corresponding DNS results (Figure 5 in [39]).

**Figure 7.**$Sh$ as a function of $Le$ from single-mode solutions (lines) with ${R}_{\rho}=0$ compared with DNS (markers) from Figure 6 and Table 3 in [39]. The horizontal wavenumbers are chosen as ${k}_{x}=\pi $, $1.25\pi $, $2\pi $, $3\pi $, $5.83\pi $ for $R{a}_{T}=50$, 100, 200, 400, 1000 based on ${k}_{x}=n\pi /{L}_{x,\mathrm{NBC}}$ with n and ${L}_{x,\mathrm{NBC}}$ as in [39].

**Figure 8.**(

**a**) One period of a standing wave computed from the single-mode equations at $R{a}_{T}=55$, ${R}_{\rho}=0.1$, $Le=5$, ${k}_{x}=\pi $ and ${k}_{y}=0$ displaying $10\left[nu\right(t)-1]$, $sh\left(t\right)$ and ${\psi}_{\mathrm{mid}}\left(t\right)$ as a function of $t\in [13.077,14.645]$ with oscillation period ${T}_{p}=1.568$. Panels (

**b**) and (

**c**) show the isocontours of the streamfunction at $t=13.077$ and $t=13.861$, respectively. This figure is to be compared with the corresponding DNS results (Figures 5 and 6 in [43]).

**Figure 9.**(

**a**) Bifurcation diagram of single-mode solutions at $R{a}_{T}=53$, $Le=5$, ${k}_{x}=\pi $ and ${k}_{y}=0$, showing steady convection rolls (—), SW (□) and TW (—). Thick lines indicate stable solutions and thin lines represent unstable solutions. (

**b**) The temporal frequency $\omega =2\pi /{T}_{p}$ of SW (□) and TW (—), the latter computed from $\omega =\left|c\right|{k}_{x}$. The Hopf frequency is ${\omega}_{\mathrm{Hopf}}=5.36981$ at the Hopf bifurcation point ${R}_{\rho}=0.10615$ (Δ) from the trivial solution. Near the termination of the SW branch, the frequency $\omega $ decreases to zero at ${R}_{\rho}^{\left(\mathrm{SW}\right)}$ as $\omega \sim 1/[-ln({R}_{\rho}-{R}_{\rho}^{\left(\mathrm{SW}\right)})]$ (−·−) as predicted theoretically [78]. Near the termination of the TW branch, the phase velocity c of the waves decreases to zero at ${R}_{\rho}^{\left(\mathrm{TW}\right)}$ as $c\sim \sqrt{{R}_{\rho}-{R}_{\rho}^{\left(\mathrm{TW}\right)}}$ (− −) as also predicted theoretically [79,80].

**Figure 10.**Top: solution profiles for steady convection rolls from the single-mode equations at $R{a}_{T}=55$ with isocontours of (

**a**) streamfunction $\psi $, (

**b**) total temperature $1-z+T$ and (

**c**) total salinity $1-z+S$. Bottom: solution profiles for a left traveling wave with $c=-1.07$ in the comoving frame from the single-mode equations at $R{a}_{T}=53$ with isocontours of (

**d**) streamfunction $\psi $, (

**e**) total temperature $1-z+T$ and (

**f**) total salinity $1-z+S$. Other parameters are ${R}_{\rho}=0.1$, $Le=5$, ${k}_{x}=\pi $ and ${k}_{y}=0$ as used in 2D DNS with periodic B.C. in the horizontal and period ${L}_{x}=2\pi /{k}_{x}=2$ (Figure 8 in [43]).

**Figure 11.**(

**a**) Bifurcation diagram of single-mode solutions at $R{a}_{T}=100$, $Le=20$, ${k}_{x}=\pi $ and ${k}_{y}=0$ showing steady convection rolls (—) of SW (□) and TW (—). Thick lines indicate stable solutions and thin lines represent unstable solutions. (

**b**) The temporal frequency $\omega =2\pi /{T}_{p}$ of SW (□) and TW (—), the latter computed from $\omega =\left|c\right|{k}_{x}$. The Hopf frequency is ${\omega}_{\mathrm{Hopf}}=23.40889$ at the Hopf bifurcation point ${R}_{\rho}=0.58548$ (Δ) from the trivial solution. Near the termination of the TW branch the phase velocity c of the waves decreases to zero at ${R}_{\rho}^{\left(\mathrm{TW}\right)}$ as $c\sim \sqrt{{R}_{\rho}-{R}_{\rho}^{\left(\mathrm{TW}\right)}}$ (− −) as predicted theoretically [79,80].

**Figure 12.**Top: unstable steady convection rolls from single-mode equations showing (

**a**) mean temperature $1-z+{\overline{T}}_{0}$ and isocontours of (

**b**) streamfunction $\psi $, (

**c**) total temperature $1-z+T$ and (

**d**) total salinity $1-z+S$. Bottom: stable left traveling wave convection in the comoving frame with phase speed $c=-5.31$ from single-mode equations showing (

**e**) mean temperature $1-z+{\overline{T}}_{0}$ and isocontours of (

**f**) streamfunction $\psi $, (

**g**) total temperature $1-z+T$ and (

**h**) total salinity $1-z+S$. The parameters are $R{a}_{T}=100$, ${R}_{\rho}=0.4$, $Le=20$, ${k}_{x}=\pi $ and ${k}_{y}=0$.

**Figure 13.**(

**a**) $Sh$ and (

**b**) $Nu$, both as a function of ${R}_{\rho}$ from the single-mode equations (lines) at $Le=20$, ${k}_{y}=0$ and $R{a}_{T}=100$, 150, 300 and 600 with wavenumbers ${k}_{x}=n\pi /{L}_{x,\mathrm{NBC}}=\pi $, $2\pi $, $2\pi $, $4\pi $, respectively, compared with the corresponding DNS results (markers) (Figure 5 in [40]).

**Figure 14.**(

**a**) $Sh$ and (

**b**) $Nu$, both as a function of $Le$, from the single-mode equations (lines) at ${R}_{\rho}=0.2$, ${k}_{y}=0$, and $R{a}_{T}=100$, 150, 300 and 600 and wavenumbers ${k}_{x}=n\pi /{L}_{x,\mathrm{NBC}}=\pi $, $2\pi $, $2\pi $, $4\pi $, respectively, compared with the corresponding DNS results (markers) (Figure 4 in [40]).

**Table 1.**Comparison of $\underset{t}{\mathrm{max}}\phantom{\rule{0.277778em}{0ex}}{\psi}_{\mathrm{mid}}\left(t\right)$, $\underset{t}{\mathrm{max}}\phantom{\rule{0.277778em}{0ex}}nu\left(t\right)$, $\underset{t}{\mathrm{max}}\phantom{\rule{0.277778em}{0ex}}sh\left(t\right)$ and the oscillation period ${T}_{p}$ of standing waves obtained from the single-mode equations and DNS (Figure 5 and p. 77 in [43]) at $R{a}_{T}=55$, ${R}_{\rho}=0.1$ and $Le=5$. The single-mode solutions are associated with ${k}_{x}=\pi $, ${k}_{y}=0$ while the DNS results [43] are computed with no-flux B.C. in a horizontal domain of size ${L}_{x,\mathrm{NBC}}=\pi /{k}_{x}=1$.

$\underset{\mathit{t}}{\mathit{max}}\phantom{\rule{0.277778em}{0ex}}{\mathit{\psi}}_{\mathit{mid}}\left(\mathit{t}\right)$ | $\underset{\mathit{t}}{\mathit{max}}\phantom{\rule{0.277778em}{0ex}}\mathit{nu}\left(\mathit{t}\right)$ | $\underset{\mathit{t}}{\mathit{max}}\phantom{\rule{0.277778em}{0ex}}\mathit{sh}\left(\mathit{t}\right)$ | Period ${\mathit{T}}_{\mathit{p}}$ | |
---|---|---|---|---|

Standing waves from DNS [43] | 0.670 | 1.052 | 1.594 | 1.535 |

Standing wave from single-mode | 0.705 | 1.058 | 1.652 | 1.568 |

**Table 2.**Comparison of ${\psi}_{\mathrm{max}}$, $Nu$, $Sh$ and c between single-mode solutions and DNS for steady convection rolls at $R{a}_{T}=55$ and traveling waves at $R{a}_{T}=53$. Other parameters are ${R}_{\rho}=0.1$, $Le=5$, ${k}_{x}=\pi $ and ${k}_{y}=0$; the DNS results are computed with periodic B.C. in the horizontal with period ${L}_{x}=2\pi /{k}_{x}=2$ (Figure 8 and p. 79 in [43]).

${\mathit{\psi}}_{\mathbf{max}}$ | $\mathit{Nu}$ | $\mathit{Sh}$ | c | |
---|---|---|---|---|

Steady convection rolls from DNS $(R{a}_{T}=55)$ [43] | 1.924 | 1.371 | 3.320 | 0 |

Steady convection rolls from single-mode $(R{a}_{T}=55)$ | 1.812 | 1.341 | 3.387 | 0 |

Traveling wave from DNS $(R{a}_{T}=53)$ [43] | 0.869 | 1.087 | 1.865 | −1.03 |

Traveling wave from single-mode $(R{a}_{T}=53)$ | 0.848 | 1.083 | 1.828 | −1.07 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Liu, C.; Knobloch, E.
Single-Mode Solutions for Convection and Double-Diffusive Convection in Porous Media. *Fluids* **2022**, *7*, 373.
https://doi.org/10.3390/fluids7120373

**AMA Style**

Liu C, Knobloch E.
Single-Mode Solutions for Convection and Double-Diffusive Convection in Porous Media. *Fluids*. 2022; 7(12):373.
https://doi.org/10.3390/fluids7120373

**Chicago/Turabian Style**

Liu, Chang, and Edgar Knobloch.
2022. "Single-Mode Solutions for Convection and Double-Diffusive Convection in Porous Media" *Fluids* 7, no. 12: 373.
https://doi.org/10.3390/fluids7120373